It turns out that having a good linear approximation is the same as being differen- tiable! You are going to prove it. Before starting, convince...

. It turns out that having a good linear approximation is the same as being differen-tiable! You are going to prove it. Before starting, convince yourself that “f has a good linear approximation near a”is equivalent to there existing a line with equation 3; = L(:E) such that: 11mm = 0. 3—H]. 27—61, (3) (a) Prove that a differentiable function always has a good linear approximation. More specifically, let a 6 1R. Let f be a function differentiable at a. Lety = L(a:) be the equation of the line tangent to the graph of y = f (3:) at thepoint with :E-coordinate a. Prove Equation (3). (b) Reciprocally, prove that if a function has a good linear approximation, thenit is differentiable. More specifically, let a 6 1R. Let f be a function that iscontinuous at a. Let y = L(r) be an arbitrary line. Assume Equation (3).Prove that f must be differentiable at a and that the slope of this line is equalto f’(a). Remember that you are trying to prove that f is differentiable at a. Do notassume it!