1873_solutions

Observe that ft f0 tt lim lim limt 0 t0 t0 t0 t t0 and

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Unformatted text preview: x | k|t x |. If t  x this inequality is obvious. Suppose that t x. Using the mean value theorem we choose a number c between t and x such that ft fx fc tx and observe that ft fx |f c | k tx 7. Given that f is a function defined on an interval S and that the inequality |f t f x | |t x | 2 holds for all numbers t and x in S, prove that f must be constant. Given any number x S we deduce from the inequality ft fx 0 |t x | tx that holds for all t x and the sandwich theorem that ft fx lim  0. tx tx Since f x  0 for every x in the interval S, the function f must be constant. S 8. Suppose that f and g are functions defined on R and that f x  g x and g x  f x for every real number x. a. Prove that f x  f x for every number x. Quite simply, for each x f x g x  fx. 242 b. Prove that the function f 2  g 2 is constant. If we define h  f 2  g 2 then h  2ff  2gg  2fg 2fg  0 and so h is constant. 9. Given that f is differentiable on the interval 0, Ý and that f x f x1 f x  as x Ý. Solution: Suppose that  0 and, using the fact that f x  as x Ý, prove that  as x Ý, choose a number w such that the inequality holds whenever x |f x  |  w. We shall now show that the inequality f x1 f x  holds whenever x w. Suppose that x w. Applying the mean value theorem to f on the interval x, x  1 we now choose a number c between x and x  1 such that f x1 f x f c x1 x and we observe that f x1 f x f x1 f x    |f c  |  . x1 x 10. Given that f is continuous on a, b and differentiable on a, b , and that f x approaches a limit w R as x a, prove that f must be differentiable at the number a and that f a  w. We need to show that ft fa w ta as t a . Suppose  0 and choose   0 such that the condition |f x w |  whenever x a, b and x  a  . Given any number t a, b satisfying the inequality a  t  a   we choose x between a and t such that ft fa f x ta and conclude that ft fa w. ta a x t a 11. Prove that if f is differentiable on an interval a, b and f a  0 and f b  0 then there must be at least one number c a, b for which f c  0. Hint: Look at the number at which the function f takes its minimum value. We know from Fermat’s theorem that f does not have its minimum at a or at b and so f must have its minimum at some number c between a and b. From Fermat’s theorem again we deduce that f c  0. 12. Prove that if f is differentiable on an interval S then the range of the function f must be an interval. Hint: Use the preceding exercise. This argument should now be an exact parallel to the Bolzano intermediate value theorem for continuous functions. 13. Suppose that f is differentiable on the interval 0, Ý , that f 0  0 and that f is increasing on 0, Ý . Prove 243 that if gx  fx x for all x  0 then the function g is increasing on 0, Ý . Solution: In order to prove that the differentiable function g is increasing on 0 for every number x  0. We therefore need to show that xf x f x 0 x2 for all x  0 and we can express this desired inequal...
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