Question

Multiple Choice Which of the following is true about the graph of \(f(x)=x^{4 / 5}\) at \(x=0 ?\) (A) It has a corner. (B) It has a cusp. (C) It has a vertical tangent. (D) It has a discontinuity. (E) \(f(0)\) does not exist.

Short answer

\nThe graph of the function \(f(x)=x^{4 / 5}\) at \(x=0\) has a vertical tangent. So, the correct answer is (C) It has a vertical tangent.

Step-by-step solution

Calculating \(f(0)\)

Firstly, the value of the function at \(x=0\) needs to be evaluated. Given that \(f(x)=x^{4 / 5}\), then \(f(0)=0^{4 / 5}\). Since any number raised to any power other than negative is 0, \(f(0)=0\).

Calculating the Derivative \(f'(x)\)

The derivative of the function demonstrates how it changes as \(x\) varies. The derivative of \(f(x)=x^{4 / 5}\) can be calculated using the power rule, resulting in \(f'(x) = \frac{4}{5} * x^{(-1 / 5)}\).

Evaluating the Derivative at \(x=0\)

Evaluating the derivative at \(x=0\) gives \(f'(0) = \frac{4}{5} * 0^{(-1 / 5)}\). Since the zero in the denominator leads to a mathematical impossibility, the derivative is undefined at \(x=0\), resulting in a vertical tangent.