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.
