Practicing Success
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 |
(a), (d) (a), (c) (b), (c) (c), (d) |
(a), (c) |
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. |