Question

A function is given. U(x) =x- x² Find the local maximum value of the function and the value of x at which it occurs. State each answer correct to two decimal place (x, y) =____

Answer 1

The local maximum value of the function U(x) = x - x² is 0.25, and it occurs at x = 1/2.

Step-by-step explanation:

To find the local maximum value of the function U(x) = x - x², we need to find the critical points by finding where the derivative of the function is equal to zero. Let's differentiate U(x) with respect to x: U'(x) = 1 - 2x. Setting U'(x) = 0, we get 1 - 2x = 0. Solving for x, we have x = 1/2. To determine if this is a local maximum, we can examine the second derivative of the function. Differentiating U'(x) = 1 - 2x with respect to x, we get U''(x) = -2. Since U''(1/2) = -2 is negative, this confirms that x = 1/2 is a local maximum value. Substituting x = 1/2 back into the original function, we can find the value of U(x): U(1/2) = 1/2 - (1/2)² = 1/2 - 1/4 = 1/4 = 0.25. Therefore, the local maximum value of the function U(x) is (1/2, 0.25).