Curve sketching

When sketching a curve, the key to locating where it bends upward or downward is the second derivative, $f^{\prime \prime }(x)$. If $f^{\prime \prime }(x)>0$ on an interval, the graph is concave up (shaped like a cup) and the slope $f^{\prime }(x)$ is increasing. If $f^{\prime \prime }(x)<0$, the graph is concave down (shaped like a cap) and the slope is decreasing. Points where $f^{\prime \prime }(x)=0$ and the sign of $f^{\prime \prime }$ changes are called inflection points; they mark a switch in concavity. To find them, differentiate twice, solve $f^{\prime \prime }(x)=0$, and test the sign of $f^{\prime \prime }$ on each side of the solutions. This single idea—using the sign of the second derivative—lets you determine all concave‑up, concave‑down, and inflection regions needed for curve sketching.