6. (This follows from Video 6.11.) Let f be a differentiable function defined on an open interval I. We will compare the following two concepts: • Recall that f is concave-up on I when f' is increasing on I. • We say that f is green when it is differentiable and "the graph of f stays above its tangent lines". This means that for every two different points P and Q on the graph of f, if Q" is a point on the line tangent to the graph of f at P, and Q' has the same x-coordinate as Q, then the y-coordinate of Q is larger than the y-coordinate of Q'. y=f(x) P Q' b (a) Show that "f is green" is equivalent to the following condition: On I f(b) - f(a) Va, bEI, a<b⇒ f'(a)< < f'(b) b-a Throditional scrant Hint: One inequality comes from assuming P is to the left of Q, and the other one from assuming P is to the right of Q. (b) Use the MVT to prove that if f is concave-up, then it is green. (c) Prove that if f is green, then it is concave-up.