Let h(x) = f(x) - (f(x))^2 + (f(x))^3 for every real number x . T… - Vidyadip

MCQ

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

✓
$h$ is increasing whenever $ f $ is increasing
B
$h$ is increasing whenever $f$ is decreasing
C
$h$ is decreasing whenever $ f $ is increasing
D
Nothing can be said in general

✓

Answer

Correct option: A.

$h$ is increasing whenever $ f $ is increasing

a
(a) $h(x) = f(x) - {[f(x)]^2} + {[f(x)]^3}$

$h'(x) = f'(x) - 2f(x)f'(x) + 3{[f(x)]^2}f'(x)$

$ = f'(x)[1 - 2f(x) + 3{[f(x)]^2}]$

$ = 3f'(x)\left\{ {{{\left( {f(x) - \frac{1}{3}} \right)}^2} + \frac{2}{9}} \right\}$

$\therefore $ $h'(x)$ and $f'(x)$ have same sign.

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