Calculus How To

Cusps in Graphs & Corners in Graphs

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A cusp or corner in a graph is a sharp turning point. These are critical points: either a local maximum (the tallest point on the graph) or local minimum (the lowest point).

Cusps in Graphs: Examples

In general, the easiest way to find cusps in graphs is to graph the function with a graphing calculator.
Example: The function f(x) = x2/3 has a cusp at x = 0. This is shown on the following graph:

cusp on a graph

A cusp is a sharp curve on a graph. Graphed with

The first derivative is undefined at x = 0 because of division by zero:
cusp on a graph nondifferentiable 2

Example #2: The graph of f(x) = |x2 – 4| has two cusps: one at x = 2 and one at x = -2:
graph with two cusps

The function isn’t differentiable at those critical points because of division by zero:
nondifferentiable function

Corners work the same way: they are sharp turns:

corner in a graph

The function f(x) = x1/3 has a sharp corner at x = 0. The slope here is undefined because it’s completely vertical (straight up and down).

Why are Functions with Cusps and Corners not Differentiable?

A function is not differentiable if it has a cusp or sharp corner. As well as the problems with division by zero shown above, we can’t even find limits near the cusp or corner because the slope to the left of the cusp is different than the slope to the right.

For example, the following graph shows f(x) = x2/3 and its derivative near the cusp:

why are cusps not differentiable

The function f(x) = x(2/3) (green) and its derivative (red).

As you approach x = 0 (the place where the cusp is) from the left, the derivative (green graph) has a different value from what you get if you approach x = 0 from the right.

Corners have the same problems: approaching from the left or from the right gives you different values for the derivative. The following graph shows a corner (a vertical tangent) for the function f(x) = x1/3 and it’s derivative (blue graph).
corner not differentiable

References graphing calculator (images).

Stephanie Glen. "Cusps in Graphs & Corners in Graphs" From Calculus for the rest of us!

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