Final answer:
The absolute maximum of the function f(x) = x + (16/x) over the interval [4,10] occurs at x = 10 and the absolute minimum occurs at x = 4.
Step-by-step explanation:
The function given is f(x) = x + (16/x), and we want to find the absolute maximum and minimum values of this function on the closed interval [4,10]. To do this, we need to calculate the function's values at the critical points and the endpoints of the interval. First, let's find the critical points by taking the derivative of the function and setting it equal to zero:
f'(x) = 1 - 16/x^2
To find where the derivative is zero, we solve 1 - 16/x^2 = 0, which gives us x = 4. However, this is also the endpoint of our interval, so this point must be checked along with the other endpoint, x = 10.
Now let's plug these x-values into the original function to find the corresponding y-values:
- f(4) = 4 + (16/4) = 8
- f(10) = 10 + (16/10) = 11.6
Comparing these values, we see that the absolute maximum value occurs at x = 10 with a value of 11.6, and the absolute minimum value occurs at x = 4 with a value of 8.
Therefore, the correct answer is:
- b. Absolute maximum occurs at x = 10
- c. Absolute minimum occurs at x = 4