`h(p) = (p - 1)/(p^2 + 4)` Find the critical numbers of the function

Textbook Question

Chapter 4, 4.1 - Problem 36 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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mathace | (Level 3) Assistant Educator

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Given: `h(p)=(p-1)/(p^2+4)`

Find the critical value(s) by setting the first derivative equal to zero and solving for the x value(s). Find the derivative by using the quotient rule.

`h'(p)=[(p^2+4)(1)-[(p-1)(2p)]]/[p^2+4]^2=0`

`h'(p)=p^2+4-(2p^2-2p)=0`

`h'(p)=p^2+4-2p^2+2p=0`

`h'(p)=-p^2+2p+4=0`

`h'(p)=p^2-2p-4=0`

`p^2-2p-4=0`

`p^2-2p+1=4+1`

`(p-1)^2=5`

`p-1=+-sqrt(5)`

` ` `p=1+-sqrt(5)`

The critical values are `x=1+sqrt(5),x=1-sqrt(5).`

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loves2learn | (Level 3) Salutatorian

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Critical values would be where there are

  1. Endpoints
  2. f’(x)=0
  3. f’(x) DNE

` h'(p) = [ 4 + 2p - p^2 ] / (p^2 + 4)^2 `

Set the numerator equal to zero as the first and third rules do not apply

`0=4+2p-p^2 `

`-4 = 2p - p^2 `
`4 = -2p + p^2 `
`5 = (-1 + p)^2 `
`1 +- sqrt(5) = p `

Therefore, the critical values are `p=1+-sqrt5 `

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