Question

Investigating for maxima & minima for the function, $f\left(x\right)={\int }_1^x[2(t-1){(t-2)}^3+3{(t-1)}^2{(t-2)}^2]dt$ we get maxima occurs at $x=k$ and minima occurs at $x=p$.Find $k+10\left(p\right)$?

Answer 1

$f\left(x\right)={\int }_1^x[2(t-1){(t-2)}^3+3{(t-1)}^2{(t-2)}^2]dt$
Using leibniz theroem,
$f^{\prime }\left(x\right)=2(x-1){(x-2)}^3+3{(x-1)}^2{(x-2)}^2=(x-1)(x-2{)}^2[2(x-2)+3(x-1)]$
$f^{\prime }\left(x\right)=(x-1)(x-2{)}^2(5x-7)=0$ for max. or min.
But at $x=2$, $f^{\prime \prime }\left(x\right)=0$ Thus $x=2$ does not corresponds to max. or min.
$\Rightarrow x=1,\frac{7}{5}$
$f\left(1\right)=0$, (local maxima) $f\left(\frac{7}{5}\right)=-\frac{108}{3125}$ (local minima)