Question

Let $f\left(x\right)$ be a polynomial function of degree $4,$ vanishes at $x=-1$. If $f\left(x\right)$ has local maxima/minima at $x=1,2,3$ and $\underset{2}{\overset{-2}{\int }}f\left(x\right)dx=\frac{1348}{15}.$ Then

• A. the $y$-intercept of the tangent to the curve at $x=-1$ equals $-96$
• B. the maximum value of $f\left(x\right)$ is $-63$
• C. $\underset{-1}{\overset{1}{\int }}[f\left(x\right)+f(-x)]dx=-\frac{1424}{15}$
• D. the minimum value of $f\left(x\right)$ is $-64$

Answer 1

Answer: (A), (D)

The correct options are
A the $y$-intercept of the tangent to the curve at $x=-1$ equals $-96$
D the minimum value of $f\left(x\right)$ is $-64$

Since, $f\left(x\right)$ be a polynomial function of degree $4,$ having $x=1,2,3$ as critical points.
$\therefore f^{\prime }\left(x\right)=A(x-1)(x-2)(x-3)$
$\Rightarrow f^{\prime }\left(x\right)=A(x^3-6x^2+11x-6)$
Integrating both the sides,
$f\left(x\right)=\frac{A}{4}(x^4-8x^3+22x^2-24x)+C$

As $f\left(x\right)=0$ at $x=-1,$ then $C=-\frac{55A}{4}$
$\Rightarrow f\left(x\right)=\frac{A}{4}(x^4-8x^3+22x^2-24x-55)$

Now, $\underset{2}{\overset{-2}{\int }}f\left(x\right)dx=\frac{1348}{15}$
$\Rightarrow \underset{-2}{\overset{2}{\int }}f\left(x\right)dx=-\frac{1348}{15}$
$\Rightarrow \frac{A}{4}\underset{-2}{\overset{2}{\int }}(x^4-8x^3+22x^2-24x-55)dx=-\frac{1348}{15}$
Using even-odd function property, we get
$\Rightarrow \frac{A}{2}\underset{0}{\overset{2}{\int }}(x^4+22x^2-55)dx=-\frac{1348}{15}$
$\Rightarrow -\frac{337}{15}A=-\frac{1348}{15}$
$\Rightarrow A=4$

Hence, $f\left(x\right)=x^4-8x^3+22x^2-24x-55$


The minimum value of $f\left(x\right)$ is obtained at $x=1,3$ and $min\left[f\right(x\left)\right]=-64.$
The maximum value of $f\left(x\right)$ is unbounded.

$\Rightarrow f^{\prime }\left(x\right)=4(x-1)(x-2)(x-3)$
$\therefore f^{\prime }(-1)=-96$ and $f(-1)=0$
Hence, equation of tangent to $y=f\left(x\right)$ at $x=-1,$ is
$y-0=-96(x+1)$
Now, for $y$-intercept, put $x=0$ we get, $y=-96$

$I=\underset{-1}{\overset{1}{\int }}[f\left(x\right)+f(-x)]dx$
$=4\underset{0}{\overset{1}{\int }}(x^4+22x^2-55)dx$
$=-\frac{2848}{15}$