Question

If $f\left(t\right)=\underset{1}{\stackrel{t}{\int }}\left(\begin{array}{c}2(x-4)(x-5{)}^3+3(x-4{)}^2(x-5{)}^2\end{array}\right)dx$, then

• A. $f\left(t\right)$ is maximum at $t=4$ and minimum at $t=\frac{22}{5}$.
• B. $f\left(t\right)$ is maximum at $t=5$ and minimum at $t=4$.
• C. $f\left(t\right)$ is minimum at $t=5$ and maximum at $t=4$.
• D. $f\left(t\right)$ is minimum at $t=5$ and maximum at $t=-20$.

Answer 1

Answer: (A)

$f\left(t\right)$ is maximum at $t=4$ and minimum at $t=\frac{22}{5}$.

$f^{\prime }\left(t\right)=2(t-4)(t-5{)}^3+3(t-4{)}^2(t-5{)}^2$

$=(t-4)(t-5{)}^2[2t-10+3t-12]$

$=-(t-4)(t-5{)}^2(5t-22)$

$-++-$

$\overline{45\frac{22}{5}}$

$f\left(x\right)$ attains min. at $x=\frac{22}{5}$

and max. at $x=4$.

Note: One may check second derivative as well.