Question

As $x$ varies from $-1$ to $+3,$ which one of the following describes the behavior of the function $f\left(x\right)=x^3-3x^2+1?$

• A. $f\left(x\right)$ increases monotonically
• B. $f\left(x\right)$ increases, then decreases and increaess again
• C. $f\left(x\right)$ decreases, then increases and decreases again
• D. $f\left(x\right)$ increases and then decreases

Answer 1

The correct option is B $f\left(x\right)$ increases, then decreases and increaess again

Given $f\left(x\right)=x^3-3x^2+1$
$f^{\prime }\left(x\right)=3x^2-6x=3x(x-2)$
Critical points are $x=0$ and $2$
while corner points are $-1$ and $3$
so sign of $f^{\prime }\left(x\right)$ is:



If we look at $f\left(x\right)$ at $xϵ[-1,+3];$ it is clear that $f\left(x\right)$ first increases then decreases and again increases.