Given f'(x)= (x-4)(4-2x) , find the x-coordinate for the relative maximum on the graph of f(x)

A) 4
B) 3
C) 2
D) none of these

To find the x-coordinate of the relative maximum on the graph of f(x), we need to look for the critical points of the function. A critical point occurs where the derivative of the function is either zero or undefined.

First, let's find the values of x for which the derivative, f'(x), is equal to zero. In this case, f'(x) = (x - 4)(4 - 2x). We set f'(x) equal to zero and solve for x:

(x - 4)(4 - 2x) = 0

Setting each factor equal to zero, we have two possibilities:

x - 4 = 0 ---> x = 4

4 - 2x = 0 ---> -2x = -4 ---> x = 2

So we have two potential critical points: x = 4 and x = 2.

Now we need to determine whether these points are relative maximum points on the graph of f(x). To do this, we can use the second derivative test. However, in order to do that, we need to find the second derivative, f''(x), first.

The second derivative is the derivative of the first derivative. Let's find f''(x):

f''(x) = (4 - 2x)' = -2

Since the second derivative, f''(x), is a constant (-2), we can't use the second derivative test to determine whether x = 4 or x = 2 correspond to relative maximum points.

Instead, we need to observe the behavior of the function around these critical points. We can analyze this by checking the sign of the first derivative on intervals between the critical points.

For x < 2, the factors (x-4) and (4-2x) are both positive, so f'(x) < 0. This suggests a decreasing portion of the curve.

For 2 < x < 4, the factor (4-2x) becomes negative, so f'(x) > 0. This suggests an increasing portion of the curve.

For x > 4, the factors (x-4) and (4-2x) are both negative, so f'(x) < 0. This suggests a decreasing portion of the curve.

Now let's summarize the behavior of f'(x) and relate it to the x-coordinate of the relative maximum:
- For x < 2 and x > 4, f'(x) < 0, indicating a decreasing slope.
- For 2 < x < 4, f'(x) > 0, indicating an increasing slope.

Based on this information, we can conclude that x = 2 corresponds to a relative maximum on the graph of f(x).

Therefore, the correct answer is C) 2.

relative max is where

f' = 0 and f" < 0
So, it will be either at x=4 or x = 2
f" = 12-4x
so, f" < 0 at x=4