Question

Find the greatest and the least values of the following functions.
$y=x^3+3x^2+3x+2$ on the interval [-2, 2].

Answer 1

$y=3x^2+6x+3$

$\frac{dy}{dx}=3(x+1{)}^2=0\Rightarrow x=-1$

$x=-1$

$\frac{d^{2}y}{d{x}^2}=6(x+1)$

${\left(\frac{d^{2}y}{d{x}^2}\right)}_{x=-1}=0$, $(-1,1)$ is the point of inflection

$f(-2)=0$ is the minimum in the given interval

$f\left(2\right)=28$ is the maximum value in the given interval