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Given the following function, find: a. zeros and the multiplicity of each b. number of turning...

Question:

Given the following function, find:

a.{eq}\; {/eq} zeros and the multiplicity of each

b.{eq}\; {/eq} number of turning points

c.{eq}\; {/eq} end behavior

{eq}f(x) = \left(x - 1\right)\left(x - 5\right) {/eq}

Turning Point and End Behaviour of a Function

Let {eq}y=f(x) {/eq} be a function then

  • {eq}x=x_0 {/eq} is a turning point of the function if the derivative {eq}f'(x_0) {/eq} changes sign at the point {eq}x=x_0 {/eq}. Usually, turning point is obtained by setting {eq}f'(x_0) = 0 {/eq}.
  • the end behaviour of the function describes the behaviour of the function at the ends of the x-axis.

Answer and Explanation:

a) We have {eq}f(x) = (x-1)(x-5) {/eq}. The zeroes are obtained by setting {eq}f(x) = 0 {/eq}. The zeroes are {eq}x = 1 {/eq} and {eq}x=5 {/eq} with each having multiplicity of one.


b) We have {eq}f(x) = (x-1)(x-5) = x^2-6x+5 \,\,\Rightarrow\,\, f'(x) = 2x-6. {/eq}

Setting {eq}f'(x) = 2x-6 = 0 \,\,\Rightarrow\,\, x=3 {/eq}.

We note that {eq}f'(x) <0 {/eq} for {eq}x<3 {/eq} and {eq}f'(x)>0 {/eq} for {eq}x>3 {/eq}.

Hence {eq}x=3 {/eq} is a turning point of the given functiion.


c) As {eq}x\to \infty {/eq}, {eq}f(x)\to \infty {/eq}.

As {eq}x\to -\infty {/eq}, {eq}f(x)\to \infty {/eq}.

Hence, the given function gets bigger and bigger as we approach the ends of the x-axis.


Learn more about this topic:

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Finding Critical Points in Calculus: Function & Graph

from CAHSEE Math Exam: Tutoring Solution

Chapter 8 / Lesson 9
196K

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