Question

The critical point(s) of the curve $y=x+\frac{3x^2}{2}+\frac{x^4}{4}+x^3$ is/are ______

Answer 1

$y=x+\frac{3x^2}{2}+\frac{x^4}{4}+x^3$
Differentiate w.r.t $x$, we get
$\frac{dy}{dx}=1+3x+x^3+3x^2=(x+1{)}^3$

As the given curve is a polynomial function, the critical points are given by,
$\frac{dy}{dx}=0\Rightarrow (x+1{)}^3=0$
$\Rightarrow x=-1$ is the critical point of the given curve.