Q. 29 - Calculus

Question

Use a sign chart for ${\mathit{f}}^{\mathbf{\text{'}}}$ to determine the intervals on which each function $\mathit{f}$ is increasing or decreasing. Then verify your algebraic answers with graphs from a calculator or graphing utility.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{x}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}}$

Step-by-Step Solution

Answer

Ans: 

Intervals of the given function are $\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{2}\mathbf{)}$

Increasing at $\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{2}\mathbf{)}$

Decreasing at $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{]}\mathbf{\cup }\mathbf{[}\mathbf{2}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Step 1. Given information:

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\frac{\mathbf{x}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}\mathbf{4}}$

Step 2. Finding the derivative of the function:

$$\begin{gathered} f\left(x\right)=\frac{x}{x^2+4} \\ {f}^{\text{'}}\left(x\right)=\frac{(x^2+4)\left(1\right)-x\left(2x\right)}{{\left(x^2+4\right)}^2} \frac{ \partial }{\partial x}\frac{u}{v}=\frac{v\frac{ \partial }{\partial x}\left(u\right)-u\frac{ \partial }{\partial x}\left(v\right)}{v^2} \\ {f}^{\text{'}}\left(x\right)=\frac{(x^2+4)-2x^2}{{\left(x^2+4\right)}^2}=\frac{-x^2+4}{{\left(x^2+4\right)}^2} \\ let f^{\text{'}}\left(x\right)=0 \\ \therefore \frac{-x^2+4}{{\left(x^2+4\right)}^2}=0 \\ \Rightarrow -x^2+4={\left(x^2+4\right)}^2 \\ \Rightarrow -x^2+4=x^4+8x^2+16 \\ \Rightarrow x^4+9x^2+12 \\ x=2,-2 \end{gathered}$$

Step 3. Inserting the root points on the number line(Sign chart):

After inserting the root values we can find the increasing and decreasing intervals of the given function. 

Intervals of the given function is $\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{2}\mathbf{)}$

Increasing at $\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{,}\mathbf{2}\mathbf{)}$

Decreasing at $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{-}\mathbf{2}\mathbf{]}\mathbf{\cup }\mathbf{[}\mathbf{2}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Step 4. Verifying algebraic answers with graphs :