SATHEE: Application of Derivatives 2 Question 19

Application of Derivatives 2 Question 19

####19. If $h (x) = f (x) - f (x)^{2} + f (x)^{3}$ for every real number $x$ . Then,

(1998, 2M)

(a) $h$ is increasing, whenever $f$ is increasing

(b) $h$ is increasing, whenever $f$ is decreasing

(d) Nothing can be said in general

Fill in the Blanks

Show Answer

Answer:

Correct Answer: 19. (b)

Solution:

Given, $h (x) = f (x) - f (x)^{2} + f (x)^{3}$

On differentiating w.r.t. $x$ , we get

$\begin{aligned} h^{'} (x) = f^{'} (x) - & 2 f (x) \cdot f^{'} (x) + 3 f^{2} (x) \cdot f^{'} (x) \\ = f^{'} (x) [1 - 2 f (x) + 3 f^{2} (x)] \\ = 3 f^{'} (x) (f (x))^{2} - \frac{2}{3} f (x) + \frac{1}{3} \\ = 3 f^{'} (x) f (x) - \frac{1}{3} + \frac{1}{3} - \frac{1}{9} \\ = 3 f^{'} (x) f (x) - {\frac{1}{3}}^{2} + \frac{3 - 1}{9} \\ = 3 f^{'} (x) f (x) - {\frac{1}{3}}^{2} + \frac{2}{9} \end{aligned}$

NOTE $h^{'} (x) < 0$ , if $f^{'} (x) < 0$ and $h^{'} (x) > 0$ , if $f^{'} (x) > 0$

Therefore, $h (x)$ is an increasing function, if $f (x)$ is increasing function and $h (x)$ is decreasing function, if $f (x)$ is decreasing function.

Therefore, options (a) and (c) are correct answers.

Application of Derivatives 2 Question 19

Fill in the Blanks

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Application of Derivatives 2 Question 19

Fill in the Blanks

Answer:

Engineering Preparation

Medical Preparation

NCERT Books & Solutions

Important Resources

Other Resources

Syllabus

Test Platform

Sample Mock Test

Mentorship

NCERT Exercise Video Solution

Menu