Question

Conside the function f(x)=x to the power 4-2xcube+2x-1

A. Find the critical points of f(x)

B. Determine the interval over which f(x) is increasing and the interval on which it is decreasing.

Answer 1

Let us consider the function $f(x)=x^4-2x^3+2x-1$.

A. Let us find the critical points of $f(x)$. The funtion is differentiable in all points of the real line. Since $f'(x)=4x^3-6x^2+2$, we conclude that $4x^3-6x^2+2=0$ implies $2(x-1)(2x^2-x-1)=0$, and hence $2(x-1)^2(2x+1)=0$. Therefore, $x_1=1$ and $x_2=-0.5$ are critical points of the function $f(x).$

B. Let us determine the interval over which $f(x)$ is increasing and the interval on which it is decreasing. Taking into account that $(x-1)^2\ge0$, we conclude that for $x>-0.5$ we have that $f'(x)=2(x-1)^2(2x+1)\ge 0$, and hence the function $f(x)$ is increasing on the interval $(-0.5, +\infty)$. For $x<-0.5$ we have that $f'(x)=2(x-1)^2(2x+1)< 0$, and hence the function $f(x)$ is decreasing on the interval $(-\infty, -0.5).$