which intervals is function f(x)=12x^(2/3)-4x increasing

Question

A) find the intervals on which f is increasing

 

 

B) Find the x and y coordinates of all relative maximum points

 

 

 

C) Find the x and y coordinates of all relative minimum points.

 

 

 

D) Find the interval on which f is concave downward

 

 

 

Answer 1

We need the first and second derivatives for this one.

 

f'(x) = 12(2/3)x^-1/3 - 4 = 8x^-1/3 - 4

 

f"(x) = -(8/3)x^-4/3

 

Observe that both derivatives are undefined at x = 0, and that the second derivative is negative everywhere it is defined.

The first derivative will be more useful if we write it as a single fraction, using the common denominator of  x^1/3. You can use algebra to verify that

 

   f'(x) = (8 - 4x^1/3)/x^1/3

 

The critical values are x=0, where the derivative is undefined, and x=8, where the derivative is 0.

 

a) If we test values of x that are less than 0, between 0 and 8, and greater than 8, we find that the function is decreasing on (-inf,0) and (8,inf), and increasing on (0, 8).

 

c) This, as well as the fact that the graph is always concave downward, tells us that there is a relative minimum at (0,0).

 

b) Since the graph is concave downward at x=8, there is a relative maximum at (8, 16).

 

d) The function is concave downward on  (-inf, 0) and (0, inf)