Q.Find the local maximum and local minimum values of $f(x) = x^4 - 8x^2 + 2$ using the second derivative test.

The second derivative test classifies a critical point $c$ (where $f'(c)=0$) using only the value of $f''(c)$, rather than the sign of $f'$ on both sides as the first derivative test requires: $f''(c)<0$ gives a local maximum, $f''(c)>0$ gives a local minimum, and $f''(c)=0$ is inconclusive, requiring a fall-back to the first derivative test. The syllabus presents this as "given as a provable tool," and its intuition is concavity -- $f''(c)<0$ means the slope $f'$ is decreasing through $c$ (consistent with a $+\to-$ sign change, hence a maximum, with the curve bending like the top of a dome), while $f''(c)>0$ means the slope is increasing through $c$ (consistent with a $-\to+$ change, hence a minimum, curve bending like the bottom of a bowl). It is usually faster than the first derivative test whenever $f''$ is easy to compute, but the most common error using it is sign-misclassification -- reading a negative second derivative as a minimum or a positive one as a maximum, exactly backwards.