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
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter2: Equations And Inequalities
Section2.6: Inequalities
Problem 80E
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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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4d0f188d-b9e0-49c3-bbf4-5bef0fd4e153%2Ffddd0dd2-4633-4389-b4f7-1534725b0ab0%2Fnus1h8k_processed.jpeg&w=3840&q=75)
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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