1. Assume y is continuous on [c, d] and differentiable on (c,d). (a) If y is non-decreasing (monotone increasing) is it true that y' > prove your answer or give a counterexample). (b) If y is strictly increasing, is it true that y' > 0? Either give a count and show where the proof of (a) fails or prove your statement.
1. Assume y is continuous on [c, d] and differentiable on (c,d). (a) If y is non-decreasing (monotone increasing) is it true that y' > prove your answer or give a counterexample). (b) If y is strictly increasing, is it true that y' > 0? Either give a count and show where the proof of (a) fails or prove your statement.
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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Following the method, prove the following.
![1. Assume y is continuous on [c, d] and differentiable on (c,d).
(a) If y is non-decreasing (monotone increasing) is it true that ✅' ≥ 0? (Either
prove your answer or give a counterexample).
(b) If y is strictly increasing, is it true that ' > 0? Either give a counterexample
and show where the proof of (a) fails or prove your statement.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F033c0d82-90fb-4c58-9b03-1325bfffdb8d%2F20967e75-fb10-4c22-9506-bb5c3e248bb6%2Fvjnety_processed.png&w=3840&q=75)
Transcribed Image Text:1. Assume y is continuous on [c, d] and differentiable on (c,d).
(a) If y is non-decreasing (monotone increasing) is it true that ✅' ≥ 0? (Either
prove your answer or give a counterexample).
(b) If y is strictly increasing, is it true that ' > 0? Either give a counterexample
and show where the proof of (a) fails or prove your statement.
![Let x, y € (a, b) and assume x <y.
By the MVT on [x,y],
fly) - f(x) = f'(c) (y_x²),
for some CE(x,y).
Now y>x a f'li) zo = fly) > f(x) for yax
The inverse to (iii) & (NV) is not true!
e.g. f(x)=x³,
8
but you ⇒ fly) > f(x).
x = [1,1] satisfies f'(x) zo (f(0) =0)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F033c0d82-90fb-4c58-9b03-1325bfffdb8d%2F20967e75-fb10-4c22-9506-bb5c3e248bb6%2Fht5n2mc_processed.png&w=3840&q=75)
Transcribed Image Text:Let x, y € (a, b) and assume x <y.
By the MVT on [x,y],
fly) - f(x) = f'(c) (y_x²),
for some CE(x,y).
Now y>x a f'li) zo = fly) > f(x) for yax
The inverse to (iii) & (NV) is not true!
e.g. f(x)=x³,
8
but you ⇒ fly) > f(x).
x = [1,1] satisfies f'(x) zo (f(0) =0)
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