Question

Consider the function f(x) = - 2x² + 102 - 6. f() has a critical point at x = A.

Find the value of A: A=
At x = A, does f(x) have a local min, a local max, or neither? Type in your answer as LMIN, LMAX, or NEITHER__________

Answer 1

The value of A is 25.5, and at x = A, the function f(x) = -2x² + 102x - 6 has a local maximum (LMAX). This is deduced from the negative coefficient of the x² term indicating a downward opening parabola.

Step-by-step explanation:

To find the critical point of the function f(x) = -2x² + 102x - 6, we first need to differentiate f(x). The derivative of f(x), denoted as f'(x), is -4x + 102. To find the critical points, we set the derivative equal to zero: -4x + 102 = 0. Solving for x, we get x = 102/4, which simplifies to x = 25.5. This means A = 25.5 is the critical point.

To determine if the critical point is a local minimum, local maximum, or neither, we check the second derivative of f(x). However, we can deduce from the negative coefficient of the x² term in the original function that the parabola opens downward, hence at x = A, f(x) has a local maximum (LMAX).