Final answer:
The local minimum y-value of the function f(x) = x⁴ − 12x² − 10 is found by determining the critical points of its derivative, which are x = 0 and x = ±√6. After evaluating these points in the original function, the local minimum y-value is -46.
Step-by-step explanation:
The function in question is f(x) = x⁴ − 12x² − 10. To find the local minimum, we need to take the derivative of the function to find the critical points where the slope is zero. The derivative of f(x) is f'(x) = 4x³ - 24x. Setting the derivative equal to zero gives us the critical points: f'(x) = 4x(x² - 6) = 0.
This implies that the critical points are at x = 0 or x = ±√6. When we substitute these values into the original function, we find the following outputs: f(0) = -10, f(√6) = -46, and f(-√6) = -46. Therefore, the local minimum y-value of the function is -46.