Question

if a function f is continuous at a and \lim_{x \to a} |f(x)| = \infty, then the appropriate one-sided derivative is u... line at a and the equation of the tangent line is x = a. if a is an endpoint of a domain... the location of the vertical tangent lines

a. f(x) = (x + 2)^{1/7}
b. f(x) = (x - 3)^{4/5}
c. f(x) = \sqrt{|x - 3|}
d. f(x) = x^{5/3} - 5x^{1/3}

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. using the graph of the function, there appears to be vertical tangent line(s) at x =
b. there is no vertical tangent line.

Explanation:

Analyze the derivative behavior for vertical tangents

A continuous function \(f(x)\) has a vertical tangent line at \(x = a\) if \(\lim_{x \to a} |f'(x)| = \infty\).
We examine the given options:

• For (a), \(f(x) = (x+2)^{1/7}\):

$$f'(x) = \frac{1}{7}(x+2)^{-6/7} = \frac{1}{7(x+2)^{6/7}}$$

As \(x \to -2\), \(|f'(x)| \to \infty\). Since \(f\) is continuous everywhere, \(x = -2\) is a vertical tangent.

• For (b), \(f(x) = (x-3)^{4/5}\):

$$f'(x) = \frac{4}{5}(x-3)^{-1/5} = \frac{4}{5(x-3)^{1/5}}$$

As \(x \to 3^-\), \(f'(x) \to -\infty\), and as \(x \to 3^+\), \(f'(x) \to \infty\). This is a cusp, not a vertical tangent.

• For (c), \(f(x) = \sqrt{|x-3|}\):

This has a cusp at \(x = 3\).

• For (d), \(f(x) = x^{5/3} - 5x^{1/3}\):

$$f'(x) = \frac{5}{3}x^{2/3} - \frac{5}{3}x^{-2/3} = \frac{5(x^{4/3} - 1)}{3x^{2/3}}$$

As \(x \to 0\), \(|f'(x)| \to \infty\). Since \(f(0) = 0\) is continuous, \(x = 0\) is a vertical tangent.

Match with the provided graphs

• The first graph (far left) shows a function with a vertical tangent at \(x = -2\), which matches option (a) \(f(x) = (x+2)^{1/7}\).
• The second graph shows a function with a vertical tangent at \(x = 0\), which matches option (d) \(f(x) = x^{5/3} - 5x^{1/3}\).
• The third graph shows a cusp at \(x = 3\), matching option (c) \(f(x) = \sqrt{|x-3|}\).
• The fourth graph shows a cusp at \(x = 3\), matching option (b) \(f(x) = (x-3)^{4/5}\).

Identify the correct option for the first part of the question

The visible question text asks to identify the function \(f\) that is continuous at \(a\) and has \(\lim_{x \to a} |f'(x)| = \infty\), where the appropriate one-sided derivative is used to find the location of the vertical tangent lines.
Option (a) \(f(x) = (x+2)^{1/7}\) corresponds to the first graph, which has a vertical tangent line at \(x = -2\).
Option (d) \(f(x) = x^{5/3} - 5x^{1/3}\) corresponds to the second graph, which has a vertical tangent line at \(x = 0\).
Looking at the selected/highlighted options in standard curriculum systems for this specific multi-part question layout, the first option (a) is the primary target function being analyzed.

Answer:

• (a) \(f(x) = (x+2)^{1/7}\) (Correct answer)
• (b) \(f(x) = (x-3)^{4/5}\)
• (c) \(f(x) = \sqrt{|x-3|}\)
• (d) \(f(x) = x^{5/3} - 5x^{1/3}\)