1. Given f (x) = x (x + 2) (x² – 5), state :a. each real zero and its' multiplicityb. whether the curve crosses or touches the x – axis at each x – interceptthe power function that the graph of f resembles for large positive andnegative values of xc.

Answered: Given f (x) = x (x + 2) (x? – 5), state…

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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1. Given f (x) = x (x + 2) (x² – 5), state :
a. each real zero and its' multiplicity
b. whether the curve crosses or touches the x – axis at each x – intercept
the power function that the graph of f resembles for large positive and
negative values of x
c.

Expert Solution

Step 1

Consider the function $f (x) = x (x + 2) (x^{2} - 5)$

(a) We have to find the real zero with multiplicity

For finding real zero equate $f (x) = 0$

$\begin{array}{l} \Rightarrow x (x + 2) (x^{2} - 5) = 0 \\ \Rightarrow x = 0 or x + 2 = 0 or x^{2} - 5 = 0 \\ \Rightarrow x = 0 or x = - 2 or x = \pm \sqrt{5} \end{array}$

Hence the real zero of the given function is $x = 0, x = - 2, x = \sqrt{5} and x = - \sqrt{5}$

And multiplicity of all zeros is $1$

Step 2

(b) Draw the graph of the function $f (x) = x (x + 2) (x^{2} - 5)$ , which is given below

Therefore, The curve crosses the $x - a x i s$ at all the zeros.

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