Maxima and Minima in Calculus | Finding maxima and minima using derivatives (2022)

Maxima and minima in calculus are found by using the concept of derivatives. As we know the concept the derivatives gives us the information regarding the gradient/ slope of the function, we locate the points where the gradient is zero and these points are called turning points/stationary points. These are points associated with the largest or smallest values (locally) of the function.

The knowledge of maxima/minima is essential to our day-to-day applied problems. Further, the article also discusses the method of finding the absolute maximum and minimum.

Maxima and minima calculus problems with solutions are given in this article.

Maxima and Minima Points

Figure for the curve with stationary points is shown below. Thus it can be seen from the figure that before the slope becomes zero it was negative, after it gets zero it becomes positive. It can be said dy/dx is -ve before stationary point dy/dx is +ve after stationary point. Hence it can be said d2 y/dx2 is positive at the stationary point shown below, Therefore it can be said wherever the double derivative is positive it is the point of minima. Vice versa wherever the double derivative is negative is negative is the point of maxima on the curve. This is also known as the second derivative test.

Maxima and Minima in Calculus | Finding maxima and minima using derivatives (1)

Let f be a function defined on an open interval I.
Let f be continuous at a critical point c in I.
If f'(x) does not change sign as x increases through c, then c is neither a point of local maxima nor a point of local minima. Such a point is called a point of inflection.

Stationary Points vs Turning Points

Stationary points are the points where the slope of the graph becomes zero. In other words the tangent of the function becomes horizontal dy/dx = 0. All the stationary points are given by the figure shown below A,B and C. And the points which the function changes its path if it was going upward it will go downward vice versa i.e. points A and B are turning points since the curve changes its path. But the point C is not turning point although the graph is flat for a short period of time but continues to go down from left to right.

Maxima and Minima in Calculus | Finding maxima and minima using derivatives (2)

Derivative Tests

Derivative test helps to find the maxima and minima of any function. Usually, first order derivative and second order derivative tests are used. Let us have a look in detail.

First Order Derivative Test

Let f be the function defined in an open interval I. And f be continuous at critical point c in I such that f’(c) = 0.

1. If f’(x) changes sign from positive to negative as x increases through point c, then c is the point of local maxima. And the f(c) is the maximum value.

2. If f’(x) changes sign from negative to positive as x increases through point c, then c is the point of local minima. And the f(c) is the minimum value.

3. If f’(x) doesn’t change sign as x increases through c, then c is neither a point of local nor a point of local maxima. It will be called the point of inflection.

Second Derivative Test

Let f be the function defined on an interval I and it is two times differentiable at c.

i. x = c will be point of local maxima if f'(c) = 0 and f”(c)<0. Then f(c) will be having local maximum value.

ii. x = c will be point of local minima if f'(c) = 0 and f”(c) > 0. Then f(c) will be having local minimum value.

iii. When both f'(c) = 0 and f”(c) = 0 the test fails. And that first derivative test will give you the value of local maxima and minima.

Properties of maxima and minima

1.If f(x) is a continuous function in its domain, then at least one maximum or one minimum should lie between equal values of f(x).

2.Maxima and minima occur alternately. I.e between two minima there is one maxima and vice versa.

3.If f(x) tends to infinity as x tends to a or b and f’(x) = 0 only for one value x i.e.c between a and b , then f(c) is the minimum and the least value. If f(x) tends to – ∞ as x tends to a or b , then f(c) is the maximum and the highest value.

Examples on Maxima and Minima

Question 1: Find the turning points of the function y = 4x3 + 12x2 + 12x + 10.

Answer: For turning points dy/dx = 0.

dy/dx = 12x2 + 24x + 12 = 0

=> 3x2 + 6x + 3 = 0

(Video) Relative Extrema, Local Maximum and Minimum, First Derivative Test, Critical Points- Calculus

=> (x + 1)(3x + 1) = 0

=> x= -1 and x = (-1)/3

Second Derivative Test:

At x = -1 :

d2 y/dx2 = 24x + 24 = 24(-1) + 24 = -24 + 24 = 0.

Hence x = -1 is the point of inflection. Therefore it is a non turning point.

At x = (-1)/3:

d2y/dx2 = 24x + 24 = 24((-1)/3) + 24 = -8 + 24 = 16.

Hence x = (-1)/3 is a point of minima. Therefore it is a turning point.

Question 2: Find the local maxima and minima of the function f(x) = 3x4 + 4x3 – 12x2 + 12.

Answer:

For stationary points f'(x) = 0.

f'(x)= 12x3 + 12x2 – 24x = 0

=> 12x(x2 + x – 2) = 0

=> 12x(x – 1)(x + 2) = 0

=> Hence, x = 0, x = 1 and x = -2

Second derivative test:

f”(x) = 36x2 + 24x – 24

f”(x) = 12(3x2 + 2x – 2)

At x = -2

f”(-2) = 12(3(-2)2 + 2(-2) – 2) = 12 (12 – 4 – 2) = 12(6) = 72 > 0

At x = 0

f”(0) = 12(3(0)2 + 2(0) – 2) = 12(-2) = -24 < 0

(Video) Finding Local Maxima and Minima by Differentiation

At x = 1

f”(1) = 12(3(1)2 + 2(1) – 2) = 12(3+2-2) = 12(3) = 36 > 0

Therefore, by the second derivative test x=0 is the point of local maxima while x = -2 and x=1 are the points of local minima.

Question 3: Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.

Answer:

Let r and h be the radius and height of the right circular cylinder inscribed in a given cone of radius R and height H. Let S be the curved surface area of cylinder.

S = 2πrh

h = H(R – r)/R

So S = 2πrH(R – r)/R

= (2πH/R)(rR – r2)

Differentiate w.r.t.r

dS/dr = (2πH/R)(R – 2r)

For maxima or minima

dS/dr =0

=> (2πH/R)(R – 2r) = 0

=> R – 2r = 0

=> R = 2r

=> r = R/2

d2S/dr2 = (2πH/R)(0 – 2)

= -4πH/R (negative)

So for r = R/2, S is maximum.

Question 4: A stone is thrown in the air. Its height at any time t is given by h = -5t2+10t+4.

Find its maximum height.

(Video) How to sketch,find maxima and minima,point of inflection using derivatives

Solution:

Given h = -5t2+10t+4

dh/dt = -10t+10

Now find when dh/dt = 0

dh/dt = 0 ⇒ -10t+10 = 0

⇒ -10t = -10

t = 10/10 = 1

Height at t = 1 is given by h = -5×12+10×1+4

= -5+10+4

= 9

Hence the maximum height is 9 m.

Question 5: Find the maxima and minima for f(x) = 2x3-21x2+36x-15

Solution:

We have f(x) = 2x3-21x2+36x-15

f’(x) = 6x2-42x+36

Now find the points where f’(x) = 0

f’(x) = 0 ⇒ 6x2-42x+36 = 0

⇒ x2-7x+6 = 0

⇒ (x-6)(x-1) = 0

⇒ x = 6 or x = 1 are the possible points of minima or maxima.

Let us test the function at each of these points.

f’’(x) = 12x-42

At x = 1, f’’(1) = 12-42 = -30 <0

(Video) Maxima and Minima

Therefore x = 1 is a point of local maximum.

The maximum value is f(1) = 2-21+36-15 = 2

At x= 6, f’’(x) = 12×6-42 = 30> 0

Therefore x = 6 is a point of local minimum.

The local minimum value is f(6) = 2(6)3-21(6)2+36(6)-15

= 2×216-21×36+216-15

= 432-756+216-15 = -123

Maxima and Minima – Top 12 Must-Do Important JEE Questions

Maxima and Minima in Calculus | Finding maxima and minima using derivatives (3)

Frequently Asked Questions

What do you mean by maxima and minima of a function?

Maxima and minima are the maximum and the minimum value of a function respectively, within the given set of ranges.

How to find maxima and minima algebraically?

We use first order derivative test and second order derivative test to find the maxima and minima of a function.

Give the steps to find maxima and minima of a function.

Let f(x) be a function. Find the first derivative dy/dx.

Then equate dy/dx to zero and find the critical points.

Then find the second derivative d2y/dx2.

Substitute the critical points in d2y/dx2.

If d2y/dx2 > 0, then that is a point of minima.

If d2y/dx2 < 0, then that is a point of maxima.

(Video) Applications of Differentiation: Rate Problems, and Maxima & Minima problems | Differential Calculus

FAQs

How do you find maxima using differentiation? ›

Must equal zero so to find them all we need to do is take the derivative. And find the zeros of the

What is the formula of maxima and minima? ›

The absolute maxima on the graph takes place at x = d, and the absolute minima of that graph takes place at x = a.

What is the condition for maxima and minima in differentiation? ›

If at a stationary point the first and possibly some of the higher derivatives vanish, then the point is or is not an extreme point, according as the first non-vanishing derivative is of even or odd order. If it is even, there is a maximum or minimum according as the derivative is negative or positive.

How do you find local min and max using first derivative? ›

Finding Local Maximum and Minimum Values of a Function - YouTube

What is maxima in derivatives? ›

A maximum is a high point and a minimum is a low point: In a smoothly changing function a maximum or minimum is always where the function flattens out (except for a saddle point).

Why is first derivative zero at maxima or minima? ›

Thus at the point of maxima or minima, one finds that the slope is zero. Hence the first derivative is zero. If the derivative is equal to zero at a certain point, it does not necessarily means that the point is a minimum of the function. ...

How do you solve the question of maxima minima? ›

How to Find Maxima and Minima Points Using Differentiation ?
  1. Differentiate the given function.
  2. let f'(x) = 0 and find critical numbers.
  3. Then find the second derivative f''(x).
  4. Apply those critical numbers in the second derivative.
  5. The function f (x) is maxima when f''(x) < 0.
  6. The function f (x) is minima when f''(x) > 0.

What is maxima and minima in algebra? ›

In an algebraic function, the maxima is the point where the value is the highest, and the minima is the point where the value is the lowest.

How many types of minima and maxima are there? ›

There are two types of maxima and minima of interest to us, Absolute maxima and minima and Local maxima and minima. Definition f has an absolute maximum or global maximum at c if f(c) ≥ f(x) for all x in D = domain of f. f(c) is called the maximum value of f on D.

How do you find the local minimum of a derivative? ›

Correct answer:

To find the local minimum of any graph, you must first take the derivative of the graph equation, set it equal to zero and solve for . To take the derivative of this equation, we must use the power rule, . We also must remember that the derivative of a constant is 0.

Is local minimum first derivative? ›

If, however, the derivative changes from negative (decreasing function) to positive (increasing function), the function has a local (relative) minimum at the critical point. When this technique is used to determine local maximum or minimum function values, it is called the First Derivative Test for Local Extrema.

What is first derivative formula? ›

Derivative by first principle refers to using algebra to find a general expression for the slope of a curve. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to. f ′ ( x ) = lim ⁡ h → 0 f ( x + h ) − f ( x ) h .

What is maxima formula? ›

Maxima: It is the point where two crests or troughs of two different wavefronts meet. The formula for maxima is given by dsinθ=mλ

What is maxima in calculus? ›

In calculus, we can find the maximum and minimum value of any function without even looking at the graph of the function. Maxima will be the highest point on the curve within the given range and minima would be the lowest point on the curve. The combination of maxima and minima is extrema.

What is the first derivative maximum value? ›

By the First Derivative Test, f has a relative maximum at x=0 and relative minima at x=−1 and x=2. If f(x0)≥f(x) for all x in an interval I, then f achieves its absolute maximum over I at x0.

What if second derivative is 0 max or min? ›

The second derivative is zero (f (x) = 0): When the second derivative is zero, it corresponds to a possible inflection point. If the second derivative changes sign around the zero (from positive to negative, or negative to positive), then the point is an inflection point.

What does 2nd derivative tell you? ›

The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to is increasing or decreasing.

Is the derivative of a maximum 0? ›

Thus, the only points at which a function can have a local maximum or minimum are points at which the derivative is zero, as in the left hand graph in figure 5.1.

What is maxima function? ›

Maxima and minima of a function are the largest and smallest value of the function respectively either within a given range or on the entire domain. Collectively they are also known as extrema of the function. The maxima and minima are the respective plurals of maximum and minimum of a function.

How do you find the maxima of a number? ›

To find the maximum number, the fractional numbers are converted to decimals and compared through the modulus operation. The first number is |7/5| = 7/5 = 1.4, the second |-3/2| = 3/2 = 1.5, and the third |-8/3| = 8/3 ≈ 2.67. From these values, we see that the maximal value is 2.67, which corresponds to -8/3 fraction.

What is application of derivatives? ›

The applications of derivatives are: To determine the rate of change of a quantity with respect to another changing quantity. To determine maximum, minimum are saddle points of a function. To determine the concavity and convexity of a function. For approximations.

What is the first derivative test? ›

First-derivative test. The first-derivative test examines a function's monotonic properties (where the function is increasing or decreasing), focusing on a particular point in its domain. If the function "switches" from increasing to decreasing at the point, then the function will achieve a highest value at that point.

What is derivative in basic calculus? ›

derivative, in mathematics, the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations.

How do you find a derivative? ›

Definition of the Derivative - YouTube

How do you find maxima and minima in partial differentiation? ›

Algorithm to find maxima and minima of two-variable functions :
  1. Find the values of x and y using fxx=0 and fyy=0 [NOTE: fxx and fyy are the partial double derivatives of the function with respect to x and y respectively.]
  2. The Obtained result will be considered as stationary/turning points for the curve.
22 Oct 2021

How do you find the range of a function by differentiation? ›

To find the range by differentiation you will have to use the concept of Maxima and minima. Whenever in a graph the value of y is maximum and minimum in the both the cases the slope is 0.

How do you find the minimum value of a differential equation? ›

If the solution y = φ(x) attains its minimum value at x = x0, then y/(x0) = 0. Hence, 2(1 + x0)(1 + y(x0)2) = 0 and thus x0 has to be −1. Check x = −1 is the minimum point of the solution y = tan(x2 + 2x).

What is the formula of partial derivatives? ›

Partial Derivative Formulas and Identities

If U = f(x,y) and both the variables x and y are differentiable of t i.e. x = g(t) and y = h(t), here we can consider differentiation as total differentiation. The total partial derivative of u with respect to t is df/dt = (∂f/∂x . dx/dt) + (∂f/∂y . dy/dt).

Why is second derivative negative for maxima? ›

The second derivative is the rate of change of the derivative, and it is negative for the process described above since the first derivative (slope) is always getting smaller. The second derivative is always negative for a "hump" in the function, corresponding to a maximum.

What is the formula of D in maxima minima of two variables? ›

Let D = f xx(a,b) f yy(a,b) - f xy 2(a,b) a) If D > 0 and f xx(a,b) > 0, then f has a relative minimum at (a,b). b) If D > 0 and f xx(a,b) < 0, then f has a relative maximum at (a,b). c) If D < 0, then f has a saddle point at (a,b).

What is derivative in basic calculus? ›

derivative, in mathematics, the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations.

How do you find the minimum value of a function? ›

You can find this minimum value by graphing the function or by using one of the two equations. If you have the equation in the form of y = ax^2 + bx + c, then you can find the minimum value using the equation min = c - b^2/4a.

What is domain and range in calculus? ›

Domain and Range. The domain of a function is the set of values that we are allowed to plug into our function. This set is the x values in a function such as f(x). The range of a function is the set of values that the function assumes. This set is the values that the function shoots out after we plug an x value in.

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