By G.E.H. Reuter

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3 Inverse Functions For each one of the following functions f restrict f to an interval so that the inverse function g is defined in an interval containing the indicated point, and find the derivative of the inverse function at that point. 1 f(x) = x 3 + 1; find g'(2). Solution. Restrict f to [0,2] because for all x E [0,2], f'(x) = 3x 2 ::::: O. Then f(O) = 1 and f(2) = 9. Thus the inverse function g: [1,9] --7 [0,2] of f is well defined and 40 III. 2 f(x) = x 2 - X + 5; find g'(7). Solution. Restrict f to [1,3] because for all x E [1,3]' f'(x) = 2x - 1 > O.

Let € > 0 and suppose Xo is irrational. Let qo E Z+ such that l/qo < Eo For each q E Z+ with q ::; qo let Sq be the set of p E Z such that The set Sq has finitely many elements. So there are only finitely many rationals with denominator::; qo which are at distance less than 1 from Xo. So we can find 8 such that all rationals in (xo - 8, Xo + 8) have denominator > qo. To be precise we let (xo,E)}. q dist(xo, Sq) = min {dist pESq Then dist(xo, Sq) > 0 because xo is irrational, so select 8 such that o < 8 < min{l, l::;q::;qo min dist(xo, Sq)}.

The minimum verifies f'(x) = 0, which is equivalent to n - L2(ai -x) = O. 4 Let f(x) = x 3 + ax 2 + bx + c where a, b, c are numbers. Show that there is a number d such that f is convex downward if x ~ d and convex upward if x ~ d. Solution. The function f" exists and f"(x) = 6x + 2a. Then for all x ~ d = -a/3, the function f is convex downward, and for all x ~ d, f is convex upward. 3 Inverse Functions 39 Prove that a function whose derivative is bounded on an interval is Lipschitz. In particular, a C 1 function on a closed interval is Lipschitz.