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1+rx ≤ (1+x) Also, show that the equality holds only if x = O. Hint: Apply the Mean Value Theorem to the = function f(x) (1 + x)r, with r > 1, on the interval [0,x], x > Oand on the interval [x,0], x > -1 to show that 1+rx ≤ (1+x). To prove that the equality holds only if x = 0 show that when 1

1+rx ≤ (1+x) Also, show that the equality holds only if x = O. Hint: Apply the Mean Value Theorem to the = function f(x) (1 + x)r, with r > 1, on the interval [0,x], x > Oand on the interval [x,0], x > -1 to show that 1+rx ≤ (1+x). To prove that the equality holds only if x = 0 show that when 1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Use the Mean value theorem to prove that, for any real number r > 1 and x > -1 1 + rx ≤ (1+x) Also, show that the equality holds only if x = O. Hint: Apply the Mean Value Theorem to the function f(x) = (1 + x)², with r > 1, on the interval [0,x], x > Oand on the interval [x,0], x > -1 to show that 1+rx ≤ (1+x). To prove that the equality holds only if x = 0, show that when 1 + rx = (1 + x)r, the statement of the mean value theorem implies (1 + c)r-¹x = x, which forces x = 0. Explain why it forces x = O.
Transcribed Image Text:Use the Mean value theorem to prove that, for any real number r > 1 and x > -1 1 + rx ≤ (1+x) Also, show that the equality holds only if x = O. Hint: Apply the Mean Value Theorem to the function f(x) = (1 + x)², with r > 1, on the interval [0,x], x > Oand on the interval [x,0], x > -1 to show that 1+rx ≤ (1+x). To prove that the equality holds only if x = 0, show that when 1 + rx = (1 + x)r, the statement of the mean value theorem implies (1 + c)r-¹x = x, which forces x = 0. Explain why it forces x = O.
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