Question

Investigate for maxima the functions $f\left(x\right)={\int }_1^x[2(t-1){(t-2)}^3+3{(t-1)}^2{(t-2)}^2]dt.$

• A. $1$
• B. $2$
• C. $3$
• D. $\frac{7}{5}$

Answer 1

The correct option is A $1$

We have
$f\left(x\right)={\int }_1^x[2(t-1){(t-2)}^3+3{(t-1)}^2{(t-2)}^2]dt.$
$={\int }_1^x(t-1){(t-2)}^2[2(t-2)+3(t-1)]dt$
$={\int }_1^x(t-1){(t-2)}^2(5t-7)dt$
$\therefore f^{\prime }\left(x\right)=(x-1){(x-2)}^2(5x-7)$
Now for max. or min. $f^{\prime }\left(x\right)=0$. This gives $x=1,7/5,2$
$f^{\prime }\left(x\right)=\frac{dy}{dx}=5\left[(x-1)(x-\frac{7}{5})\right]{(x-2)}^2$
By change of sign rule at $x=1$ and 7/5, we observe that y is max. at $x=1$
and min. at $x=\frac{7}{5}$, at $x=2,\frac{dy}{dx}$ does not change sign hence it is neither max. nor min. at $x=2.$