Suppose that p(t) is a differentiable function with $'(1) S kø(1) (k > 0) for 1 2 a. Multiply both sides by e-kt, then transpose to show that for t 2 a. Then apply the mean value theorem to conclude that $(1) < ¢(a)ek(t-a) for t2 a.

Suppose that p(t) is a differentiable function with $'(1) S kø(1) (k > 0) for 1 2 a. Multiply both sides by e-kt, then transpose to show that for t 2 a. Then apply the mean value theorem to conclude that $(1) < ¢(a)ek(t-a) for t2 a.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.5: The Kernel And Range Of A Linear Transformation

Problem 30EQ

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Question

Suppose that p(t) is a differentiable function with
$'(1) S kø(1)
(k > 0)
for 1 2 a. Multiply both sides by e-kt, then transpose to
show that
for t 2 a. Then apply the mean value theorem to conclude
that
$(1) < ¢(a)ek(t-a)
for t2 a.

Expert Solution

Step 1

Given: Suppose $ϕ (t)$ is a differentiable function with $ϕ' (t) \leq k ϕ (t)$ (k>0) such that $\frac{d}{d t} [ϕ (t) e^{- k t}] \leq 0$ .

To Prove: for $t \geq a$ , $ϕ (t) \leq ϕ (a) e^{(t - a) k}$ .

Step by step

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