Question

Consider the function f(x) = x 4 −2kx2 . Here k is a positive constant. At what value(s) of x does f(x) obtain is absolute minima

Answer 1

To find the absolute minima of the function f(x) = x^4 - 2kx^2, we need to find the critical points of the function and determine whether they are relative minima. The critical points are x = 0 and x = ±√(k), and the second derivative is used to determine that only x = ±√(k) represent stable equilibria and absolute minima.

Step-by-step explanation:

To find the absolute minima of the function f(x) = x4 - 2kx2, we need to find the critical points of the function. The critical points are the values of x where the derivative of the function is equal to zero or undefined.

First, let's find the derivative of f(x) using the power rule: f'(x) = 4x3 - 4kx.

Setting f'(x) = 0, we can solve for x: 4x3 - 4kx = 0. Factoring out a common factor of 4x, we get x(4x2 - 4k) = 0. This equation has two solutions: x = 0 and x = ±√(k).

Now, we need to determine whether these critical points are relative minima. We can do this by examining the second derivative of f(x). The second derivative is f''(x) = 12x2 - 4k.

For x = 0, f''(0) = -4k. Since k is a positive constant, f''(0) is negative. Therefore, x = 0 represents a relative maximum and not an absolute minimum.

For x = ±√(k), f''(±√(k)) = 12(√(k))2 - 4k = 8k - 4k = 4k. Since k is positive, f''(±√(k)) is positive. Therefore, x = ±√(k) represents relative minima and are the values where f(x) obtains its absolute minima.