Question

For what values of x is $f\left(x\right)=-x^3+3x^2-2x+2$ concave or convex ?

Answer 1

$f\left(x\right)=-x^3+3x^2-2x+2$

$f^{\prime }\left(x\right)=-3x^2+6x-2$

$f^{\prime \prime }\left(x\right)=-6x+6$

Now, $f^{\prime \prime }\left(x\right)=⟹x=1$

Then, we consider a number larger than 1 and a number smaller than 1 and substitute the values in our second derivative.

If number is greater than 1, $f^{\prime \prime }\left(x\right)<0$

If number is less than 1, $f^{\prime \prime }\left(x\right)>0$

Hence, $f\left(x\right)$ is concave up in the interval $(-\infty ,1)$ and concave down on the interval $(1,\infty )$.