Answer:
*The function has a minimum in x=-1
*The function has a maximum in x=1
*The second derivative is not enough to determine if the function has either a maximum or a minimum in x=0.
Step-by-step explanation:
1. Evaluate the second derivative in the first critical point x=-1:
[tex]h"(x)=x^{3}-4x^{2}+5x[/tex]
[tex]h"(-1)=(-1)^{3}-4(-1^{2})+5(-1)[/tex]
[tex]h"(-1)=-1-4-5[/tex]
[tex]h"(-1)=-10[/tex]
As the value is smaller than zero, the function has a minimum in x=-1
2. Evaluate the second derivative in the second critical point x=1
[tex]h"(x)=x^{3}-4x^{2}+5x[/tex]
[tex]h"(1)=(1)^{3}-4(1^{2})+5(1)[/tex]
[tex]h"(1)=1-4+5[/tex]
[tex]h"(1)=2[/tex]
As the value is larger than zero, the function has a maximum in x=1
3. Evaluate the second derivative in the third critical point x=0
[tex]h"(x)=x^{3}-4x^{2}+5x[/tex]
[tex]h"(0)=(0)^{3}-4(0^{2})+5(0)[/tex]
[tex]h"(0)=0-0+0[/tex]
[tex]h"(0)=0[/tex]
As the value is equal to zero, the second derivative is not enough to determine if the function has either a maximum or a minimum in x=0.
