Question

Consider the polynomial $f\left(x\right)=1+2x+3x^2+4x^3$. Let $s$ be the sum of all distinct real roots of $f\left(x\right)$ and let $t=|s|$, then the function $f^{\prime }\left(x\right)$ is?

• A. increasing in $(-t,-\frac{1}{4})$ and decreasing in $(-\frac{1}{4},t)$
• B. decreasing in $(-t,-\frac{1}{4})$ and increasing in $(-\frac{1}{4},t)$
• C. increasing in $(-t,t)$
• D. decreasing in $(-t,t)$

Answer 1

Answer: (B)

decreasing in $(-t,-\frac{1}{4})$ and increasing in $(-\frac{1}{4},t)$

Given, $f\left(x\right)=1+2x+3x^2+4x^3$

$f^{\prime }\left(x\right)=2(1+3x+6x^2)$
$f"\left(x\right)\Rightarrow 12x+3$
$12x+3\Rightarrow 0$
$\underset{\text{critical point}}{\underset{\uparrow }{x}}=\frac{-1}{4}(\because x=t\to$ only has one root)
$f^{\prime }\left(x\right)$ in $x<\frac{-1}{4}$ is less than '$0$' hence decreasing function.
$f^{\prime }\left(x\right)$ in $x>\frac{-1}{4}$ is more than '$0$' hence increasing function.