Question

Consider the function f(x) = 5x + 5x⁻¹ . For this function there are four important intervals: (- [infinity], A] , [A, B); (B, C) and [C, [infinity]) where A, and C are the critical numbers and the function is not defined at B. Find A= ___

Answer 1

The critical numbers of the function are A = 1 and B = 0.

Step-by-step explanation:

The function f(x) = 5x + 5x⁻¹ has four important intervals: (-∞, A], [A, B), (B, C), and [C, ∞), where A and C are the critical numbers and the function is not defined at B. We need to find the value of A. Since the function is not defined at B, we know that B is equal to 0, as B represents the point where the denominator becomes zero, resulting in an undefined value. To find A, we need to determine the critical numbers of the function, which occur when the derivative of f(x) equals zero or is undefined. Taking the derivative of f(x), we get f'(x) = 5 - 5x⁻². Setting this equal to zero, we have 5 - 5x⁻² = 0. Solving for x, we find x⁻² = 1, which implies x = 1. Therefore, the critical numbers are A = 1 and B = 0.