At what values of x does f(x) = x³ − x have a local maximum or minimum?

Solution:

Maxima and minima of functions are useful techniques to calculate optimal solutions to many problems. These can be found out by drawing graphs or using differentiation. However, the method of derivatives is more convenient since many functions have very complex graphs which are very difficult to plot. Let's solve an example related to the concept. Let's understand the solution in detail.

The function f(x) = x³ − x is a cubic function.

By using differentiation, we can find its local maxima and minima.

Step 1: We differentiate once and equate the first derivative of the function to zero.

⇒ f'(x) = 3x² − 1 = 0

⇒x = 1/√3, -1/√3

Step 2: We found the optima using the first derivative. Now we find the nature of the optima using the second derivative.

⇒ f''(x) = 6x

When we find f''(x) for x = 1/√3, we get f''(x) = 2√3. Hence, as f''(1/√3) is positive, we get a minima at x = 1/√3.

Similarly, when we find f''(x) for x = -1/√3, we get f''(x) = -2√3. Hence, as f''(-1/√3) is negative, we get a maxima at x = -1/√3.

Hence, the values of x at which f(x) = x³ − x have a local maximum is x = -1/√3 and a local minimum at x = 1/√3.