Respuesta :
Answer:
a) DNE
b) The function increases for every real value of x.
c) DNE
Step-by-step explanation:
Given a function f(x), the critical points are those in which [tex]f^{\prime}(x) = 0[/tex], that is, the roots of the first derivative of f(x).
Those critical points let us find the intervals in which the function increases or decreases. If the first derivative in the interval is positive, the function increases in the interval. If it is negative, the function decreases.
If the function increases before a critical point and then, as it passes the critical point, it starts to decrease, we have that the critical point [tex](x_{c}, f(x_{c}) is a relative maximum.
If the function decreases before a critical point and then, as it passes the critical point, it starts to increase, we have that the critical point [tex](x_{c}, f(x_{c}) is a relative minimum.
If the function has no critical points, it either always increases or always decreases.
In this exercise, we have that:
[tex]f(x) = 7x + 1[/tex]
(a) Find the critical numbers of f.
[tex]f^{\prime}(x) = 0[/tex]
[tex]f^{/prime}(x) = 7[/tex]
7 = 0 is false. This means that f has no critical points.
(b) Find the open intervals on which the function is increasing or decreasing.
Since there are no critical points, we know that either the function increases or decreases in the entire real interval.
We have a first order function in the following format:
[tex]f(x) = ax + b[/tex]
In which [tex]a > 0[/tex].
So the function increases for every real value of x.
(c) Apply the First Derivative Test to identify the relative extremum.
From a), we find that there are no critical numbers. So DNE
