Chapter 3.2, Problem 12E

To determine

Whether the given function $y$ is a corner, a cusp, a vertical tangent, or a discontinuity.

Answer to Problem 12E

The given function $y$ is a cusp.

Explanation of Solution

Given information:

The given function is $y=x^{4/5}$ and which is fails to be differentiable at $x=0$ .

Formula Used:

Differentiable function formula:

The differentiable function formulas to find the left hand derivative and right hand derivative are

Left hand derivative $=\underset{x\to 0^{-}}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$ and Right hand derivative $=\underset{x\to 0^{+}}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$ respectively.

Calculation:

The given function is:

  $y=x^{4/5}$

To find the Left hand derivative, use the differentiable function formula. $\text{Left}\text{hand}\text{derivative}=\underset{x\to 0^{-}}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$ .

Substitute $x^{4/5}$ for $f\left(x\right)$ and $0$ for $a$ in the above formula.

  $\begin{array}{c}\text{Left}\text{hand}\text{derivative}=\underset{x\to 0^{-}}{\lim }\frac{x^{4/5}-f\left(0\right)}{x-0}\\ =\underset{x\to 0^{-}}{\lim }\frac{x^{4/5}-0}{x}\\ =\underset{x\to 0^{-}}{\lim }\frac{x^{4/5}}{x}\\ =\underset{x\to 0^{-}}{\lim }\frac{1}{x^{1/5}}\end{array}$

Simplify the above limit further.

  $\underset{x\to 0^{-}}{\lim }\frac{1}{x^{1/5}}=-\infty$

Again, to find the Right hand derivative, use the differentiable formula $\text{Right}\text{hand}\text{derivative}=\underset{x\to 0^{+}}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$ .

Substitute $x^{4/5}$ for $f\left(x\right)$ and $0$ for $a$ in the above formula.

  $\begin{array}{c}\text{Right}\text{hand}\text{derivative}=\underset{x\to 0^{+}}{\lim }\frac{x^{4/5}-f\left(0\right)}{x-0}\\ =\underset{x\to 0^{+}}{\lim }\frac{x^{4/5}-0}{x}\\ =\underset{x\to 0^{+}}{\lim }\frac{x^{4/5}}{x}\\ =\underset{x\to 0^{+}}{\lim }\frac{1}{x^{1/5}}\end{array}$

Simplify the above limit further.

  $\underset{x\to 0^{+}}{\lim }\frac{1}{x^{1/5}}=\infty$

Thus, the results of the limit indicate that the given function is a cusp.

The graph of the function $y=x^{4/5}$ has a cusp at $x=0$

  

Hence, the given function $y=x^{4/5}$ is a cusp.