Suppose f is a function with continuous second derivative such that f(x) →∞ as x →∞ andf(x) → - as x → -o.Assume that the only solution to f" (x) = 0 is x = 2 and the only solutions to f'(x) = 0 areI =1 and x = 3.Which of the following need NOT be true:f is decreasing for 1< x <3 and is increasing otherwisef is concave down for T < 2 and concave up for x > 2f is increasing for r < 2 and decreasing for > 2f has a unique local maximum at r = 1 that is not an absolute maximum

Given that f→∞ as x→∞ and f→-∞ as x→-∞ and the solution for f''=0 is x=2 and the only solution for…

Answered: Assume that the only solution to f" (x)…

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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Suppose f is a function with continuous second derivative such that f(x) →∞ as x →∞ and
f(x) → - as x → -o.
Assume that the only solution to f" (x) = 0 is x = 2 and the only solutions to f'(x) = 0 are
I =1 and x = 3.
Which of the following need NOT be true:
f is decreasing for 1< x <3 and is increasing otherwise
f is concave down for T < 2 and concave up for x > 2
f is increasing for r < 2 and decreasing for > 2
f has a unique local maximum at r = 1 that is not an absolute maximum

Expert Solution

Step 1

Given that $f \to \infty$ as $x \to \infty$ and $f \to - \infty$ as $x \to - \infty$ and the solution for $f'' = 0$ is $x = 2$ and the only solution for $f' (x)$ is $x = 1$ and $x = 3$ .

From the information we have $f' (x) = (x - 1) (x - 3)$ and also $f'' = 2 x - 4$ satisfies all the properties from these

$\begin{array}{rcl} f & = & \int (x - 1) (x - 3) \\ = & \int x^{2} - 4 x + 3 \\ = & \frac{x^{3}}{3} - 2 x^{2} + 3 x + c \end{array}$

Without loss of generality let's assume c=0

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