Question

The locally linear approximation of the differentiable function f at x=3 is used to approximate the value of f(3. 2). The approximation at x=3. 2 is an overestimate of the corresponding function value at x=3. 2. Which of the following could be the graph of f ?

Answer 1

The linear approximation to f(x) at x = 3 is

L(x) = f(3) + f'(3) (x - 3)

so that

f(3.2) ≈ L(3.2) = f(3) + f'(3) (3.2 - 3)

f(3.2) ≈ f(3) + 0.2 f'(3)

The approximation is an overestimate, which means

f(3.2) - f(3) > 0

so that

f(3.2) ≈ f(3) + 0.2 f'(3) ⇒ f'(3) ≈ (f(3.2) - f(3))/0.2 > 0

and this in turn mean f(x) is increasing as x grows larger than 3. Pick the graph that shows an increasing function.