MATA31 - Assignment 7 (Written)

Due Date: Wednesday, November 13, 2024 by 11:59 PM.

Question 1    (10 points)

Let f : R → R be a function with domain R whose graph passes through the point (0, 0) and suppose f is continuous at 0. Define g(x) = xf(x) for all x ∈ R.

(a) Explain why the product rule may not apply to find g ′ (0).

(b) Prove g ′ (0) = 0.

Question 2    (10 points)

Let f(x) = − x + 1/1. Find all points P on the line y = x with the following property:

There is exactly one tangent line to the graph of y = f(x) that passes through the point P. Justify your answer.