Question

question 2
true or false, the above graph is differentiable at all points
true
false

Explanation:

Step1: Recall differentiability condition

A function is differentiable at a point if the left - hand and right - hand derivatives exist and are equal. For the function $y = \sqrt[3]{x}=x^{\frac{1}{3}}$, its derivative using the power rule $y'=\frac{d}{dx}(x^n)=nx^{n - 1}$ is $y'=\frac{1}{3}x^{-\frac{2}{3}}=\frac{1}{3x^{\frac{2}{3}}}$.

Step2: Analyze derivative at $x = 0$

As $x
ightarrow0$, $\lim_{x
ightarrow0^{+}}\frac{1}{3x^{\frac{2}{3}}}=\infty$ and $\lim_{x
ightarrow0^{-}}\frac{1}{3x^{\frac{2}{3}}}=\infty$. The function $y = \sqrt[3]{x}$ has a vertical tangent at $x = 0$. At a vertical tangent, the derivative is undefined.

Answer:

False