Question

Let \(f : \mathbb{R} \to \mathbb{R}\) be given by \(f(x) = |x^2 - 1|\), then:

• A. \(f\) has a local minima at \(x = 1\) but no local maxima
• B. \(f\) has a local maxima at \(x = 0\), but no local minima
• C. \(f\) has a local minima at \(x = \pm 1\) and a local maxima at \(x = 0\)
• D. \(f\) has neither any local maxima nor any local minima

Hint:

For absolute value functions, break the function into cases depending on the sign of the expression inside the absolute value. This will help you analyze the function’s behavior more easily and identify points of local maxima and minima.