Q4/ In each of the following, sketch the graph of a function with all the statedproperties:f(0) = 2, f(2) = 0, f'(0) = f'(2) = 0,f'(x) > 0 for x < 0 or x > 2f'(x) < 0 for 0 < x < 2f"(x) < 0 for x <1f"(x) > 0 for x > 1

Answered: Q4/ In each of the following, sketch…

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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Q4/ In each of the following, sketch the graph of a function with all the stated
properties:
f(0) = 2, f(2) = 0, f'(0) = f'(2) = 0,
f'(x) > 0 for x < 0 or x > 2
f'(x) < 0 for 0 < x < 2
f"(x) < 0 for x <1
f"(x) > 0 for x > 1

Expert Solution

Step 1

It is given that $f (x)$ is a real valued function with the following properties:

$f (0) = 2, f (2) = 0, f' (0) = f' (2) = 0 f' (x) > 0 f o r x < 0 o r x > 2 f' (x) < 0 f o r a < x < 2 f'' (x) < 0 f o r x < 1 f'' (x) > 0 f o r x > 1$

Since, $f'' (x) < 0$ for $x < 1$ and $f'' (x) > 0$ for $x > 1$ , we can consider

$f'' (x) = a (x - 1)$ , where $a$ is any positive arbitrary constant to be determined.

Integrating both sides, we obtain

$\int d (f' (x)) = a \int (x - 1) d x \Rightarrow f' (x) = \frac{a}{2} \{x (x - 2)\} + b, [b = i n t e g r a t i n g c o n s \tan t] (i)$

Step 2

Now, given $f' (0) = f' (2) = 0$ which on applying in eq.(i) gives $b = 0$ . Therefore,

$f' (x) = \frac{a}{2} x (x - 2)$

Clearly $f' (x) < 0$ for $x > 2$ and $f' (x) < 0$ for $0 < x < 2$ . Now

$f' (x) = \frac{1}{2} (x^{2} - x) \Rightarrow d (f (x)) = \frac{a}{2} (x^{2} - x) d x \Rightarrow \int d (f (x)) = \int \frac{a}{2} (x^{2} - x) d x \Rightarrow f (x) = \frac{a}{2} (\frac{x^{3}}{3} - x^{2}) + c, c = i n t e g r a t i n g c o n s \tan t$

Now, given $f (0) = 2, f (2) = 0$ . Using these conditions, we get

$a = 3, c = 2$

Hence the required function is

$f (x) = \frac{3 x^{2}}{2} (\frac{x}{3} - 1) + 2$

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