Respuesta :
Answer:
D)
Step-by-step explanation:
Remember, if a is critical point of f then f'(a)=0. And criterion of the second derivative says that if a is a critical point of f and
1. if [tex]f''(a)<0[/tex] then f has a relative maximum in (a,f(a)),
2. if [tex]f''(a)>0[/tex] then f has a relative minimum in (a,f(a)),
3. if [tex]f''(a)=0,[/tex] Then the criterion does not decide. That is, f may have a relative maximum at a, a relative minimum at (a, f (a)) or neither.
Since [tex]f'(5)=0[/tex] then 5 is a critical point of f. Now we apply the second criterium:
since [tex]f''(5)=0[/tex] then the criterium doesn't decide, that means, more information is needed to determine if f has a maximum or minimum at x = 5.
Using the concept of critical point and the second derivative test, it is found that the correct option is:
D. More information is needed to determine if f has a maximum or minimum at x = 5.
- The critical points of a function [tex]f(x)[/tex] are the values of x for which [tex]f^{\prime}(x) = 0[/tex].
Applying the second derivative test, we have that:
- If positive, that is, [tex]f^{\prime\prime}(x) > 0[/tex], it is a relative minimum.
- If negative, that is, [tex]f^{\prime\prime}(x) < 0[/tex], it is a relative maximum.
- If zero, that is, [tex]f^{\prime\prime}(x) = 0[/tex], we do not have sufficient information.
In this problem:
- [tex]f^{\prime}(5) = 0[/tex], thus, at x = 5 is a critical value.
- [tex]f^{\prime\prime}(5) = 0[/tex], thus, we need more information, which means that the correct option is:
D. More information is needed to determine if f has a maximum or minimum at x = 5.
A similar problem is given at https://brainly.com/question/16944025
