Question

What is true about the graph of f(x) = |x| at the point (0,0)?

Choose the correct answer below.
A. The graph has infinitely many tangent lines.
B. The graph has two tangent lines y = -x and y = x
c. The graph has no tangent line.
D. The graph has a tangent line at y = 0.

Answer 1

The graph of f(x) = |x| has an interesting feature at (0,0) where it cannot be assigned a single tangent line owing to its V-shape, suggesting two possible tangent lines with different slopes on either side, making answer B correct.

Step-by-step explanation:

The graph of the function f(x) = |x| at the point (0,0) is indeed a point of interest. Typically, a function has a single tangent line at each point on its graph. However, due to the absolute value, f(x) forms a V-shaped graph, which is not differentiable at the vertex point (0,0). This is because the left-hand side of the function approaches with a slope of -1, while the right-hand side approaches with a slope of +1. Therefore, the graph does have two distinct lines that could act as tangent lines on either side but it has no single defined tangent line at the point (0,0). Answer B is the most accurate description: The graph has two tangent lines y = -x and y = x.