Chapter 3, Problem 1RE

Finding Critical numbers In Exercises 1–6, find the critical numbers of the function.

$f\left(x\right)=-x^2-5x+9$

To determine

To calculate: The critical numbers of the function $f\left(x\right)=-x^2-5x+9$.

Answer to Problem 1RE

Solution:

The critical number of the function $f\left(x\right)=-x^2-5x+9$ is $\underset{\_}{-\frac{5}{2}}$.

Explanation of Solution

Given Information:

The provided function.

$f\left(x\right)=-x^2-5x+9$

Formula used:

To find the critical numbers of $f$, find all $x$-values for which $f^{\prime }\left(x\right)=0$ and $x$-values for which $f^{\prime }\left(x\right)$ is undefined.

Calculation:

Consider the provided function.

$f\left(x\right)=-x^2-5x+9$

The critical points for the functions is determined by taking derivative of the function.

$\begin{array}{c}{f}^{\prime }\left(x\right)=\frac{d}{dx}\left(-x^2-5x+9\right)\\ =-2x-5\end{array}$

Equate $f^{\prime }\left(x\right)=0$, as follows.

$\begin{array}{c}{f}^{\prime }\left(0\right)=-2x-5\\ -2x-5=0\\ -2x=5\\ x=-\frac{5}{2}\end{array}$

Hence, the critical number of the function $f\left(x\right)=-x^2-5x+9$ is $-\frac{5}{2}$.