find the critical numbers of the function: h(p)=p-1/p^2+4
did you mean it the way you typed it or did you mean
h(p) = p - 1/(p^2 + 4) ?
To find the critical numbers of the function h(p) = p - 1 / (p^2 + 4), we need to start by finding the derivative of the function. The derivative will help us identify where the function is changing and locate any potential critical points.
Step 1: Find the derivative of h(p)
To find the derivative, we can use the quotient rule. The quotient rule states that if we have a function f(p) = g(p) / k(p), then the derivative of f(p) is given by (g'(p) * k(p) - g(p) * k'(p)) / (k(p))^2.
In our case, g(p) = p - 1, and k(p) = p^2 + 4. Let's calculate the derivative:
h'(p) = [(g'(p) * k(p) - g(p) * k'(p)) / (k(p))^2]
= [(1 * (p^2 + 4) - (p - 1) * (2p)) / (p^2 + 4)^2]
= [(p^2 + 4 - 2p^2 + 2p) / (p^2 + 4)^2]
= [(-p^2 + 2p + 4) / (p^2 + 4)^2]
Step 2: Set the derivative equal to zero to find critical points
To find critical points, we need to set the derivative h'(p) equal to zero and solve for p:
-p^2 + 2p + 4 = 0
We can rearrange the equation to solve for p by adding p^2 - 2p - 4 to both sides:
p^2 - 2p - 4 = 0
Step 3: Solve for p using factoring or quadratic formula
To solve the quadratic equation, we can either try to factor it or use the quadratic formula. In this case, factoring may not be straightforward, so let's use the quadratic formula:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation p^2 - 2p - 4 = 0, a = 1, b = -2, and c = -4:
p = (2 ± √((-2)^2 - 4(1)(-4))) / (2(1))
p = (2 ± √(4 + 16)) / 2
p = (2 ± √20) / 2
p = (2 ± 2√5) / 2
We can simplify this expression further by factoring out a 2 from the numerator:
p = 2(1 ± √5) / 2
The 2 in the numerator can be canceled out with the 2 in the denominator:
p = 1 ± √5
So, the critical numbers of the function h(p) = p - 1 / (p^2 + 4) are p = 1 + √5 and p = 1 - √5.