Final answer:
To find the local minimum value of g(x) = x^4 - 5x^2 + 4, we calculate the derivative, set it to zero, and solve for x to find the critical points. The critical points indicate where local minima may occur, and we evaluate the function at these points to find the value.
Step-by-step explanation:
To find the local minimum value of the function g(x) = x^4 - 5x^2 + 4, we need to calculate the critical points of the function where the derivative equals zero or is undefined. The critical points occur where g'(x) = 0.
The first step is to find the derivative of g(x), which is g'(x) = 4x^3 - 10x. We then set the derivative equal to zero to find the critical points: 4x^3 - 10x = 0. This can be factored to x(4x^2 - 10) = 0, giving us critical points at x = 0 and x = ±√(5/2).
Since the leading coefficient of g(x) is positive, we know the end behavior of the function is up, implying the local minima would occur at the smallest second derivative point. By testing these critical points, we can determine that x = ±√(5/2) are indeed local minima.
We then evaluate g(x) at x = √(5/2) to find the local minimum value: g(√(5/2)) = (√(5/2))^4 - 5(√(5/2))^2 + 4. After calculations, we approximate this to the nearest hundredth.
The answer will be one of the provided options closest to our calculation.