Michael A. answered • 10/09/16

Patient, Friendly 4th and 5th Grade Math Tutor

To find the gradient, you need to take the derivative of the function h(x). Here we will need to apply the Quotient Rule.

h'(x) = [(x + 3)(2) - (2x - 5)(1)]/(x + 3)²

= (2x + 6 - 2x + 5)/(x + 3)²

= 11/(x + 3)²

First, we see that the numerator is a positive constant, and the denominator can never be negative because it is raised to the second power. This algebraically demonstrates that the function h(x) never has a negative gradient.

To find when the gradient is equal to 1, set h'(x) = 1 and solve for x.

11/(x + 3)² = 1

Cross-multiply to obtain:

(x + 3)² = 11

Expand the left side to get

x² + 6x + 9 = 11

x² + 6x - 2 = 0.

You will need to use the Quadratic Formula to solve for x. Can you handle it from here, Rikah?