XA and x & f(x)}. Show Cim ƒ.) 38. Let a, b, c, and d be any real numbers such that a < b and c < d. Prove that [a, b] is equivalent to [c, d]. (Hint: Show that [a, b] is equivalent to [0, 1] first.) 0.5 REAL NUMBERS *39. If x < y, prove that x<* + y 2

Question 22 of 22 > Find the critical points and the intervals on which the function f(x) = (x³ – 4 x) e* + 15 is increasing or decreasing. Use the First Derivative Test to determine whether the critical point is a local minimum or maximum (or neither). (Use symbolic notation and fractions where needed. Give your answer in the form of a comma separated list. Enter DNE if there are no critical points.) the critical numbers with local minimum: the critical numbers with local maximum: (Use symbolic notation and fractions where needed. Give your answers as intervals in the form (*, *). Use the symbol co for infinity, U for combining intervals, and an appropriate type of parenthesis "(",")", "[","]" depending on whether the interval is open or closed.)

44.

Use the Intermediate Value Theorem to prove that for every real x > 0 and every integer n > 0, there is a unique positive real number y such that y^n =x.

Z is the set of integers, R is the set of real numbers. 5) Functions f: R -> R, g: R -> R are both one to one on the setof real numbers R. Is the function f + g also one to one?Prove your answer.

5. (10 points) Prove that for all positive real numbers x and y, one has the inequality +2 > 2 nool numbers satisfying (x- 3)2 + (y +1)2 < 4.

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