To solve this problem, we’ll go through it step-by-step.

Given Information

The function $f(x) = \sqrt[3]{x}$ and the derivative formula for $f'(a)$: $f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$

Step 1: Find $f'(a)$ for $a \neq 0$

For $a \neq 0$, we can calculate $f'(a)$ by using the limit definition of the derivative.

1. Substitute $f(x) = \sqrt[3]{x}$ into the formula for $f'(a)$: [ f(a + h) = \sqrt[3]{a + h} \quad \text{and} \quad f(a) = \sqrt[3]{a} ]

   So, we get: [ f'(a) = \lim_{h \to 0} \frac{\sqrt[3]{a + h} - \sqrt[3]{a}}{h} ]

2. To evaluate this limit, let’s use the fact that the derivative of $\sqrt[3]{x}$ with respect to $x$ is $\frac{1}{3}x^{-\frac{2}{3}}$. Using this directly: $f'(x) = \frac{1}{3}x^{-\frac{2}{3}}$

   So, for $f'(a)$ when $a \neq 0$: $f'(a) = \frac{1}{3}a^{-\frac{2}{3}}$

Step 2: Show that $f'(0)$ does not exist and that $f$ has a vertical tangent line at $(0,0)$

To investigate the existence of $f'(0)$, we use the definition of the derivative at $x = 0$: [ f'(0) = \lim_{h \to 0} \frac{f(0 + h) - f(0)}{h} = \lim_{h \to 0} \frac{\sqrt[3]{h} - 0}{h} = \lim_{h \to 0} \frac{\sqrt[3]{h}}{h} ]

Now, rewrite $\sqrt[3]{h} = h^{\frac{1}{3}}$: $f'(0) = \lim_{h \to 0} \frac{h^{\frac{1}{3}}}{h} = \lim_{h \to 0} h^{\frac{1}{3} - 1} = \lim_{h \to 0} h^{-\frac{2}{3}}$

Since $h^{-\frac{2}{3}}$ approaches infinity as $h \to 0$, the limit does not exist. This implies that $f'(0)$ does not exist.

Conclusion: Vertical Tangent at $(0,0)$

The fact that $f'(0)$ does not exist and the derivative approaches infinity as $h \to 0$ from either direction indicates that the graph of $f(x) = \sqrt[3]{x}$ has a vertical tangent line at $(0, 0)$.

Would you like further details on any of these steps?

Related Questions

1. How does one interpret a vertical tangent geometrically on a graph?
2. How can we find the derivative of other roots, such as $\sqrt[4]{x}$?
3. What does it mean for a derivative to be infinite at a certain point?
4. How would we apply this limit method for other non-integer power functions?
5. What would the tangent line look like if $f(x) = x^{2/3}$?

Tip

When dealing with non-integer powers, converting terms to fractional exponents often simplifies the differentiation process.