The Second Derivative Test

The proof runs through the First Derivative Test. Suppose $f''(c) > 0$. Since $f''$ is continuous, it stays positive on some open interval around $c$, so $f'$ is increasing there. Since $f'(c) = 0$, $f'$ is negative just to the left of $c$ and positive just to the right. By the First Derivative Test, $f$ has a relative minimum at $c$. When $f''(c) < 0$, the same argument with the signs reversed gives a relative maximum.

Near $c$, a graph that's concave up with a horizontal tangent lies above that tangent line: a valley. A graph that's concave down at a horizontal tangent lies below it: a peak.

The test needs $f'(c) = 0$. At a critical point where $f'(c)$ doesn't exist, $f''(c)$ doesn't exist either, so use the First Derivative Test there.

Each function below has $f'(0) = 0$ and $f''(0) = 0$, and the three behave differently at $0$.

For $x^2 + 2\cos x$, $f''(x) = 2 - 2\cos x$ is positive on both sides of $0$. So $f'(x) = 2x - 2\sin x$ increases through $0$, from negative to positive, and $f$ has a relative minimum.

For $\sin^2 x - x^2$, $f''(x) = -4\sin^2 x$ is negative on both sides of $0$. So $f'(x) = \sin 2x - 2x$ decreases through $0$, and $f$ has a relative maximum.

For $\sin x - x$, $f'(x) = \cos x - 1$ is negative on both sides of $0$, so $f$ decreases through $0$ and has neither.

So when $f''(c) = 0$, go back to the First Derivative Test and find the sign of $f'$ on each side of $c$.

Here's the reason for a minimum. Suppose some $d$ in $I$, say with $d > c$, had $f(d) < f(c)$. By the Extreme Value Theorem, $f$ has a maximum on $[c, d]$. It isn't at $d$, where $f$ is below $f(c)$. If it were at $c$, $f$ would be constant just to the right of $c$, and every point there would be critical. So it's strictly between $c$ and $d$, a second critical point. The case $d < c$ is the same.

The theorem needs exactly one critical point. In Example 1, $f$ has two, and its relative maximum $f(1) = -9$ isn't an absolute maximum on $x > 0$: $f(8) = 64 - 80 + 8\ln 8 \approx 0.64$ is larger.