Let h(x) = f(x) - {f(x)}2 + {f(x)}3 for every real number x. Then
(a) h is increasing whenever f is increasing
(b) h is increasing whenever f is decreasing
(c) h is decreasing whenever f is decreasing
(d) nothing can be said in general
Answer & explanation
Correct answer: option 2
We have,
$h(x)=f(x)-\{f(x)\}^2+\{f(x)\}^3$
∴ $h'(x)=f'(x)-2 f(x) f'(x)+3\{f(x)\}^2 f'(x)$
$\Rightarrow h'(x)=f'(x)\left[1-2 f(x)+3\{f(x)\}^2\right]$
$\Rightarrow h'(x)=f'(x)\left(3 y^2-2 y+1\right)$, where y = f(x)
Consider the quadratic expression $3 y^2-2 y+1$. Clearly, discriminant of this quadratic expression is less than zero. So, its sign is always same as that of coefficient of $y^2$ i.e. positive.
∴ $h'(x)=f'(x) \times A$ positive real number.
⇒ Sign of h'(x) is same as that of f'(x)
⇒ either h'(x) > 0 and f'(x) > 0 or h'(x) < 0 and
f'(x) < 0
⇒ h(x) and f(x) increase and decrease together.