What is the difference between a local max and an inflection point

Consider a function $f(x)$. A point $x=c$ is a local maximum if there exists an interval $(c-\delta,c+\delta)$ such that $f(c)\ge f(x)$ for every $x$ in that interval. For differentiable functions the first derivative satisfies $f'(c)=0$ (or does not exist) and the second derivative is negative, $f''(c)<0$. Example: $f(x)=-x^{3}+3x$ has $f'(x)=-3x^{2}+3$; setting this to zero gives $x=1$ and $x=-1$. Evaluating $f''(x)=-6x$ shows $f''(1)=-6<0$, so $x=1$ is a local maximum with value $f(1)=2$.

An inflection point occurs where the concavity of the graph changes, meaning the second derivative passes through zero and switches sign. The first derivative may be zero, but it is not required. Using the same function $f(x)=-x^{3}+3x$, the second derivative $f''(x)=-6x$ is zero at $x=0$. To the left of zero, $f''(x)>0$ (concave up); to the right, $f''(x)<0$ (concave down). Hence $x=0$ is an inflection point, even though $f'(0)=3\neq0$.

The first‑derivative test tells whether a critical point is a maximum, minimum, or neither by checking the sign change of $f'$. The second‑derivative test is quicker: if $f''(c)<0$ the point is a local maximum, if $f''(c)>0$ a local minimum, and if $f''(c)=0$ the test is inconclusive, often indicating a possible inflection point. Therefore a zero second derivative does not guarantee a max or min; it signals a change in curvature.

To sketch a curve accurately, locate all critical points where $f'(x)=0$ or undefined, then apply the second‑derivative test to classify each as a max, min, or inconclusive. Next, find where $f''(x)=0$ or undefined to identify potential inflection points, and verify a sign change in $f''$. Finally, plot the points with their classifications; this reveals the overall shape and helps avoid misreading the graph.

Derivative tests at a glance

Procedure to classify a critical point

1. 1Compute $f'(x)$ and solve $f'(x)=0$ for critical points.
2. 2Compute $f''(x)$ at each critical point.
3. 3If $f''(c)<0$ label as local maximum; if $f''(c)>0$ label as local minimum; if $f''(c)=0$ move to next step.
4. 4Solve $f''(x)=0$ and test the sign of $f''$ on each side to confirm inflection points.

Key indicators for each type

• Local maximum: $f'(c)=0$ and $f''(c)<0$.
• Inflection point: $f''(c)=0$ and sign of $f''$ changes.
• First derivative zero does not guarantee a max/min.
• Concavity switches from up to down or down to up at an inflection.

Comparison of properties