Science:Math Exam Resources/Courses/MATH102/December 2017/Question 11 (a)/Solution 1

Let us first compute the derivatives:

$$f(x)=x^{5}-4x^{2}+2x,$$

$$f'(x)=5x^{4}-8x+2,$$

$$f''(x)=20x^{3}-8.$$

If ${\textstyle x}$ is near ${\textstyle 1}$, then ${\textstyle f''(x)}$ is close to

$$f''(1)=20\cdot 1^{3}-8=12>0.$$

Therefore, by the second derivative test, the critical is a local minimum.

Answer: The critical point is a ${\textstyle {\color {blue}{\mbox{local minimum}}}}$.