Q.The curve $y=ax^4+bx^2$ with $ab>0$

Concave up / concave down. A graph is concave up (convex down) at a point if the tangent line there lies below the graph nearby; it is concave down (convex up) if the tangent lies above the graph nearby.

Definition via $f'$. On an open interval $I$ where $f''$ exists: $f$ is concave up on $I$ if $f'$ is strictly increasing on $I$; concave down if $f'$ is strictly decreasing on $I$.

Test of Concavity (Theorem 7.11 — the working tool).

• $f''(x)>0$ on an open interval $I$ $\Rightarrow$ $f$ is concave up on $I$.
• $f''(x)<0$ on an open interval $I$ $\Rightarrow$ $f$ is concave down on $I$.

Remarks. A local maximum of a function that is convex-up (concave-down) on the whole interval $[a,b]$ is automatically its absolute maximum there (and symmetrically for a concave-up function's local minimum). There is only ever one absolute maximum (and one absolute minimum) on an interval, but there can be several local maxima/minima.

Points of inflection. A point where the graph switches from concave-up to concave-down (or vice versa) is a point of inflection.

Test for Points of Inflection (Theorem 7.12). If $f''(c)$ exists and changes sign as $x$ passes through $c$, then $\big(c,f(c)\big)$ is a point of inflection; and if $f''$ exists at a genuine point of inflection, then necessarily $f''(c)=0$ there.

Three cautions (all illustrated in the textbook remarks) that keep this test from being mechanically misapplied:

1. $f''(c)$ may fail to exist at a genuine inflection point (e.g. $f(x)=x^{1/3}$ at $x=0$) — so "$f''$ undefined" is not automatically "no inflection point," it needs separate checking. …