1. **State the problem:** We are given the function $f(x) = 2x^3$ and asked which statement about the behavior of $f$ at the point $(0,0)$ is true. 2. **Recall the definitions:** - A **corner** occurs where the function is continuous but the derivative from the left and right do not match. - A function **does not have a derivative** at a point if the derivative is undefined there. - A **horizontal tangent line** means the derivative at that point is zero. - A **vertical tangent line** means the derivative tends to infinity or is undefined due to vertical slope. 3. **Find the derivative:** $$f'(x) = \frac{d}{dx}(2x^3) = 6x^2$$ 4. **Evaluate the derivative at $x=0$:** $$f'(0) = 6 \times 0^2 = 0$$ 5. **Interpret the result:** - Since $f'(0)$ exists and equals 0, the function is differentiable at 0. - The derivative is continuous and smooth, so no corner exists at $(0,0)$. - The tangent line at $(0,0)$ is horizontal because the slope is 0. - The tangent line is not vertical because the derivative is finite. **Final answer:** $f$ has a horizontal tangent line at $(0,0)$.