Question

Let $f(x)=\underset{0}{\overset{x}{\int }}{e}^{-t^2}(t-\alpha {)}^3\left(t^2-7t+12\right)dt$ for all $x\in (0,\infty )$, if $f$ has a local maximum at $x=$ 4 and a local minimum at $x=3$ and $\alpha \in [0,10]$, then

• A. number of possible integral value of $\alpha$ is 6 .
• B. $x=\alpha$ is a point of local minimum.
• C. there exists atleast two $c_1,c_2\in (0,\infty )$ such that $f^{\prime \prime }\left(c_1\right)=0$ and $f^{\prime \prime }\left(c_2\right)=0$.
• D. there exists some $c\in (0,\infty )$ such that $f^{\prime \prime \prime }(c)=0$.

Answer 1

Answer: (A), (B), (C), (D)

Correct answer is (a) number of possible integral value of $\alpha$ is 6 . Correct answer is (b) $x=\alpha$ is a point of local minimum. Correct answer is (c) there exists atleast two $c_1,c_2\in (0,\infty )$ such that $f^{\prime \prime }\left(c_1\right)=0$ and $f^{\prime \prime }\left(c_2\right)=0$. Correct answer is (d) there exists some $c\in (0,\infty )$ such that $f^{\prime \prime \prime }(c)=0$.