Application of Derivatives 4 Question 38
40. Which of the following is true?
(a) $f(x)$ is decreasing on $(-1,1)$ and has a local minimum at $x=1$
(b) $f(x)$ is increasing on $(-1,1)$ and has a local maximum at $x=1$
(c) $f(x)$ is increasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$
(d) $f(x)$ is decreasing on $(-1,1)$ but has neither a local maximum nor a local minimum at $x=1$
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Answer:
Correct Answer: 40. $(2,1)$
Solution:
- When $x \in(-1,1)$,
$$ x^{2}<1 \Rightarrow x^{2}-1<0 $$
$\therefore f^{\prime}(x)<0, f(x)$ is decreasing.
Also, at $\quad x=1, f^{\prime \prime}(1)=\frac{4 a}{(a+2)^{2}}>0 \quad[\because 0<a<2]$
So, $f(x)$ has local minimum at $x=1$.
