Find the absolute maximum value and the absolute minimum value, if any, of the function. (If an answer does not exist, enter DNE.)

Question

g(x) = x² + 2x^2/3 on [−7, 8]

Answer 1

This function is continuous everywhere.

The derivative, 2x + 2/3x^-1/3, is positive on (0, infinity) and negative on (-infinity, 0), the function is monotone increasing on (0, infinity) and monotone decreasing on (-infinity, 0);thus, the minimum is at 0. As the function is symmetric (replace x with -x and you get the same value), we thus know that, due to being monotone on each of the intervals and symmetric, the maximum on any closed interval will be at the endpoint with the greatest absolute value. |8| = 8; |7| = 7. 8 > 7

The maximum function is at x = 8. The maximum value is 8² + 8^2/3 = 64 + (2³)^2/3 = 64 + 2² = 64 + 4 = 68