1873_solutions

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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 x1 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 x1 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 x1 f x f c x1 x and we observe that f x1 f x f x1 f x    |f c  |  . x1 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...
View Full Document

Ask a homework question - tutors are online