Question

Find the critical point and the interval on which the given function is increasing or decreasing, and apply the First Derivative Test to the critical point. Let f(x)=4 ln (6 x)-x, x>0

Answer 1

To find the critical point and interval of a given function, we can find the derivative, set it equal to zero, analyze the sign of the derivative, and apply the First Derivative Test. For the function f(x) = 4ln(6x) - x, the critical point is x = 4, the function is increasing for x > 4 and decreasing for x < 4, and the critical point is a local minimum.

Step-by-step explanation:

To find the critical point and the interval on which the given function is increasing or decreasing, we need to find the derivative of the function and set it equal to zero. Let's first find the derivative of f(x) = 4ln(6x) - x.

f'(x) = 4(1/x) - 1 = 4/x - 1.

Next, we set f'(x) equal to zero and solve for x:

4/x - 1 = 0

4/x = 1

x = 4

The critical point is x = 4. To determine the interval on which the function is increasing or decreasing, we can analyze the sign of the derivative. Since f'(x) = 4/x - 1, f'(x) will be positive for x > 4 and negative for x < 4. Therefore, the function is increasing for x > 4 and decreasing for x < 4.

Finally, we can apply the First Derivative Test to the critical point x = 4. Since the function is increasing to the right of x = 4 and decreasing to the left of x = 4, the critical point x = 4 is a local minimum.