Suppose that f is a positive increasing function on an interval I. prove that I/

Question

Suppose that f is a positive increasing function on an interval I. prove that I/f is decreasing on I.

Explanation / Answer

f is positive increasing function on I => f(x) > 0 for all x in I and f(x1) - f(x2) >= 0 whenever x1, x2 belong to I and x1 - x2 > 0 ..... (*) Now consider the function g = 1/f on I , g(x) = 1/f(x) > 0 for all x in I. now let x1, x2 belong to I and x1 - x2 > 0 , then g(x1) - g(x2) = 1/f(x1) - 1/f(x2) = [f(x2) - f(x1)] / f(x1)f(x2) from (*) it is clear that the numerator is non-positive, and the denominators are positive quantities, so overall, g(x1) - g(x2) < = 0, for any x1, x2 belonging to I and x1 - x2 > 0 . And thus g = 1/f is a decreasing function. Have a nice day !!

Related Questions

Navigate

• Browse (All)
• Browse S
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.