Question

Consider The Function F(X)=Ln(X)X5. For This Function There Are Two Important Intervals: (A,B] And [B,[infinity]) Where A And B Are Critical Numbers where the function is undefined.

Find A
Find B
For each of the following intervals tell whether f(x) is increasing (type in INC) or decreasing (type in Dec)
(A,B]:
[B,~):

Answer 1

The function f(x) = ln(x)x^5 has two important intervals (A,B] and [B,∞) where A is 0 and B is infinity. In both intervals, the function f(x) is increasing.

Step-by-step explanation:

The function f(x) = ln(x)x^5 has two important intervals: (A,B] and [B,∞) where A and B are critical numbers where the function is undefined.

To find A, we need to find the x-value where ln(x) is undefined. The natural logarithm function, ln(x), is only defined for x > 0. So A = 0.

To find B, we need to find the x-value where the function x^5 is undefined. This function is defined for all real numbers, so B = ∞.

Now, for the intervals (A,B] and [B,∞):

In the interval (A,B], the function f(x) is increasing, because both ln(x) and x^5 are increasing functions.

In the interval [B,∞), the function f(x) is also increasing, because both ln(x) and x^5 are still increasing functions.