Question

If $f'(x) = g(x)(x - a)^{2},$ where $g(a) \neq 0$ and $g$ is continuous at x = a then

• A. $f$is increasing near a if g(1)> 0
• B. f is increasing near a if g(1)< 0
• C. f is decreasing near a if g(1)>0
• D. f is decreasing near a if g(1)< 0

Answer 1

Answer: (D)

f is decreasing near a if g(1)< 0

Answer 2

Since g is continuous at x = a if g(1) >0 there exist an open interval I containing a so that $g(x) > 0$, $\forall x \in \text{ I}$

⇒ $f'(x) \geq - \forall x \in \text{ I}$.

Therefore, f is increasing near a.

Similarly f is decreasing near a

if $g(a)$ < 0.