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?