Find the points of local maxima or minima of f x = 1 x 2 + 2 using the first derivative test. Also, find the maximum and minimum values.

Given that f x = 1 x 2 + 2 Taking differentiation with respect to x we get f ' x = x 2 + 2 0 - 2 x 1 x 2 + 2 2 ∵ d d x u v = u d v d x - v d u d x v 2 = - 2 x x 2 + 2 2 For critical points f ' x = 0 ⇒ - 2 x = 0 ⇒ x = 0 ∴ x = 0 is critical point. Now, for x < 0 , f ' x > 0 , and for x > 0 , f ' x < 0 At x = 0 , f ' x changes sign from positive to negative. So, x = 0 is point of maxima. And, Maximum value is f 0 = 1 2

Find the points of local maxima or minima of f x = x 1 - x , x > 0 using the first derivation test. Also, find the local maximum or local minimum values.

Given that f x = x 1 - x , x > 0 Taking differentiation with respect to x we get f ' x = x 2 1 - x - 1 + 1 - x [Since d d x u v = u d v d x + v d u d x ] = - x + 1 - x × 2 2 1 - x = - x + 2 - 2 x 2 1 - x = 2 - 3 x 2 1 - x For Critical Points, we have f ' x = 0 ⇒ 2 - 3 x 2 1 - x = 0 ⇒ 2 - 3 x = 0 ⇒ x = 2 3 Therefore, x = 2 3 is critical point. Now, f ' x = 3 2 3 - x 2 1 - x Now, for x < 2 3 , f ' x > 0 , and for x > 2 3 , f ' x < 0 Thus, f ' x changes sign from positive to negative. ∴ x = 2 3 is point of local maxima. And, Maximum value of f x is f 2 3 = 2 3 1 - 2 3 = 2 3 × 1 3 = 2 3 3 = 2 3 9 .

Find the points of local maxima or local minima of f x = x 2 + 2 x , x > 0 using the first derivative test. Also, find the local maximum or minimum values.

Given that f x = x 2 + 2 x Taking differentiation with respect to x , we get f ' x = 1 2 - 2 x 2 [Since d d x x n = n x n - 1 , n ∈ R ] For critical points f ' x = 0 ⇒ 1 2 - 2 x 2 = 0 ⇒ x 2 - 4 2 x 2 = 0 ⇒ x 2 - 4 = 0 ⇒ x 2 = 4 ⇒ x = ± 2 Since f ' x = x - 2 x + 2 2 x 2 Now, for x < 2 , we have f ' x < 0 , and for x > 2 , we have f ' x > 0 Thus, at x = 2 , f ' x changes sign from negative to positive. ∴ x = 2 is point of minima. And, Minimum value is f 2 = 1 + 1 = 2 . Now, for x < - 2 , f ' x > 0 , and for x > - 2 , f ' x < 0 Thus, at x = - 2 , f ' x changes sign from positive to negative. ∴ x = - 2 is a point of maxima And, the maximum value of f x is f - 2 = - 2 2 + 2 - 2 = - 1 - 1 = - 2

E M B I B E

MATHEMATICS CLASS XII VOLUME-1>Chapter 18 - Maxima and Minima>EXERCISE

R D Sharma Solutions for Chapter: Maxima and Minima, Exercise 1: EXERCISE

Author:R D Sharma

R D Sharma Mathematics Solutions for Exercise - R D Sharma Solutions for Chapter: Maxima and Minima, Exercise 1: EXERCISE

Attempt the free practice questions on Chapter 18: Maxima and Minima, Exercise 1: EXERCISE with hints and solutions to strengthen your understanding. MATHEMATICS CLASS XII VOLUME-1 solutions are prepared by Experienced Embibe Experts.

Questions from R D Sharma Solutions for Chapter: Maxima and Minima, Exercise 1: EXERCISE with Hints & Solutions

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 18 - Maxima and Minima>EXERCISE>Q 2

Find the points of local maxima or minima for $f (x) = x^{3} - 3 x$ using the first derivative test. Also, find the local maximum or minimum value.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 18 - Maxima and Minima>EXERCISE>Q 3

Find the points of local maxima or minima of $f (x) = x^{3} {(x - 1)}^{2}$ using the first derivative test. Also, find the local maximum or local minimum values.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 18 - Maxima and Minima>EXERCISE>Q 4

Find the point of local maxima or minima for $f (x) = (x - 1) {(x + 2)}^{2}$ using the first derivative test. Also, find the maximum and minimum value.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 18 - Maxima and Minima>EXERCISE>Q 5

Find the points of local maxima or minima of $f (x) = \frac{1}{x^{2} + 2}$ using the first derivative test. Also, find the maximum and minimum values.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 18 - Maxima and Minima>EXERCISE>Q 12

Find the points of local maxima or minima of $f (x) = x \sqrt{1 - x}, x > 0$ using the first derivation test. Also, find the local maximum or local minimum values.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 18 - Maxima and Minima>EXERCISE>Q 13

Find the points of local maxima or minima of $f (x) = x^{3} {(2 x - 1)}^{3}$ using the first derivative test. Also, find the local maximum or local minimum values.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 18 - Maxima and Minima>EXERCISE>Q 14

Find the points of local maxima or local minima of $f (x) = \frac{x}{2} + \frac{2}{x}, x > 0$ using the first derivative test. Also, find the local maximum or minimum values.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-1>Chapter 18 - Maxima and Minima>EXERCISE>Q 10

Find the points of local maxima or local minima, if any, of the following functions, using the first derivative test. Also, find the local maximum or local minimum values, as the case may be:

$f (x) = \sin 2 x - x, - \frac{π}{2} \leq x \leq \frac{π}{2}$