Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)
h(p) = (p − 2)/(p^2 + 9)
I keep on coming up with 2 + and - squareroot(2)all over 2. Please help.
To find the critical numbers of the function h(p) = (p - 2)/(p^2 + 9), we need to find the values of p where the derivative of h(p) is either zero or undefined.
First, let's find the derivative of h(p). Using the quotient rule, we have:
h'(p) = [(p^2 + 9)(1) - (p - 2)(2p)] / (p^2 + 9)^2
= (p^2 + 9 - 2p^2 + 4p) / (p^2 + 9)^2
= (-p^2 + 4p + 9) / (p^2 + 9)^2
Next, we set the derivative equal to zero and solve for p:
-p^2 + 4p + 9 = 0
This quadratic equation does not factor easily, so we can use the quadratic formula:
p = (-4 ± sqrt(4^2 - 4(-1)(9))) / (2(-1))
= (-4 ± sqrt(16 + 36)) / (-2)
= (-4 ± sqrt(52)) / (-2)
= (-4 ± 2sqrt(13)) / (-2)
= 2 ± sqrt(13)
Thus, the critical numbers of the function h(p) = (p - 2)/(p^2 + 9) are:
p = 2 + sqrt(13) and p = 2 - sqrt(13)
Therefore, the critical numbers are 2 + sqrt(13) and 2 - sqrt(13).
To find the critical numbers of the function h(p) = (p − 2)/(p^2 + 9), we need to find the values of p where the derivative of h(p) is either zero or undefined.
Let's start by finding the derivative of h(p) using the quotient rule:
h'(p) = [((p^2 + 9)(1) - (p − 2)(2p))/(p^2 + 9)^2]
Simplifying the numerator:
h'(p) = [(p^2 + 9 - 2p^2 + 4p)/(p^2 + 9)^2]
= [(-p^2 + 4p + 9)/(p^2 + 9)^2]
Next, we set h'(p) to zero and solve for p to find the values where the derivative is zero:
-p^2 + 4p + 9 = 0
Unfortunately, this polynomial does not factor nicely. Therefore, we can use the quadratic formula to solve for p:
p = [-4 ± √(4^2 - 4(-1)(9))]/(2(-1))
= [-4 ± √(64)]/(-2)
= [-4 ± 8]/(-2)
= [-4 + 8]/(-2) or [-4 - 8]/(-2)
= 4/-2 or -12/-2
= -2 or 6
So, we have two critical numbers: p = -2 and p = 6.
Therefore, the critical numbers of the function h(p) = (p − 2)/(p^2 + 9) are -2 and 6.