Sure! Here's a step-by-step guide to find the requested values:
a) To find h'(x), we need to differentiate the function h(x) = kx³ - 4x² - 5x with respect to x. The derivative of a function is found by applying the power rule.
So, h'(x) = 3kx² - 8x - 5.
b) Next, we are given that h'(1) = 8. To find the value of k, we substitute x = 1 into the derivative function h'(x) and set it equal to 8:
3k(1)² - 8(1) - 5 = 8
3k - 8 - 5 = 8
3k - 13 = 8
3k = 21
k = 7
Therefore, the value of k is 7 when h'(1) = 8.
