Question

Find the critical numbers of the function.

30. \(h(p) = \frac{{p - 1}}{{{p^2} + 4}}\).

Short answer

The critical number of the function \(h(p)\) are \(p = 1 + \sqrt 5 \) and \(p = 1 + \sqrt 5 \).

Step-by-step solution

Given data

The given function is \(h(p) = \frac{{p - 1}}{{{p^2} + 4}}\).

Concept of Differentiation

Differentiation is a method of finding the derivative of a function. Differentiation is a process, where we find the instantaneous rate of change in function based on one of its variables.

Determine the derivative of the function

Obtain the first derivative of the given function.

Obtain the first derivative of the given function: \({h^\prime }(p) = \frac{d}{{dp}}\left( {\frac{{p - 1}}{{{p^2} + 4}}} \right)\)

Apply the Quotient rule as shown in equation,

\(\begin{aligned}{c}{h^\prime }(p) = \frac{{\left( {{p^2} + 4} \right)(1) - (p - 1)(2p)}}{{{{\left( {{p^2} + 4} \right)}^2}}}\\{h^\prime }(p) = \frac{{{p^2} + 4 - 2{p^2} + 2p}}{{{{\left( {{p^2} + 4} \right)}^2}}}\\{h^\prime }(p) = \frac{{ - {p^2} + 2p + 4}}{{{{\left( {{p^2} + 4} \right)}^2}}}\\{h^\prime }(p) = \frac{{ - \left( {{p^2} - 2p - 4} \right)}}{{{{\left( {{p^2} + 4} \right)}^2}}}\end{aligned}\)

Take \({h^\prime }(p) = 0\) and obtain the critical number.

\(\begin{aligned}{c}\frac{{\left( {{p^2} - 2p - 4} \right)}}{{{{\left( {{p^2} + 4} \right)}^2}}} = 0\\{p^2} - 2p - 4 = 0\end{aligned}\)

Obtain the roots of the equation

Obtain the roots of the equation by using the quadratic formula.

\(\begin{aligned}{l}p = \frac{{2 \pm \sqrt {4 + 16} }}{2}\\p = \frac{{2 \pm \sqrt {20} }}{2}\\p = \frac{{2 \pm 2\sqrt 5 }}{2}\\p = \frac{{2(1 \pm \sqrt 5 )}}{2}\end{aligned}\)

Therefore, the roots are, \(p = 1 + \sqrt 5 \) and \(p = 1 + \sqrt 5 \). Since the expression \({p^2} + 4\) in the denominator of \({h^\prime }(p)\) never equal to zero, \({h^\prime }(p)\) exists for all real numbers. Hence, it fails to satisfy the definition of critical number. Thus, the critical numbers of the function \(h(p)\) are \(p = 1 + \sqrt 5 \) and \(p = 1 + \sqrt 5 \).