`h(p) = (p - 1)/(p^2 + 4)` Find the critical numbers of the function
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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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Critical values would be where there are
- Endpoints
- f’(x)=0
- 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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