How do you find the max and min for #f(x)= x - ((64x)/(x+4))# on the interval [0,13]?

Question

How do you find the max and min for #f(x)=   x -  ((64x)/(x+4))# on the interval [0,13]?

William Abdalla · Accepted Answer

Have a look:

Christopher Allers · Answer

undefined To find the maximum and minimum values of the function $ f(x) = x - \frac{64x}{x+4} $ on the interval $[0, 13]$, follow these steps:

1. Find the critical points of $ f(x) $ on the interval $[0, 13]$ by setting the derivative equal to zero and solving for $ x $.
2. Evaluate the function at the critical points and at the endpoints of the interval.
3. Determine which of these values correspond to the maximum and minimum values of the function.

Let's go through these steps:

1. Calculate the derivative of $ f(x) $:
   $$ f'(x) = 1 - \frac{64(x+4) - 64x}{(x+4)^2} $$

2. Set $ f'(x) = 0 $ and solve for $ x $ to find the critical points:
   $$ 1 - \frac{64(x+4) - 64x}{(x+4)^2} = 0 $$
   Solve this equation for $ x $ to find the critical points.

3. Evaluate $ f(x) $ at the critical points obtained in step 2, and at the endpoints of the interval $[0, 13]$, which are $ x = 0 $ and $ x = 13 $.

4. Compare the values of $ f(x) $ at these points to determine the maximum and minimum values on the interval $[0, 13]$.