Q. 30 - 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.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{ }\frac{\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{1}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{1}}$

Step-by-Step Solution

Answer

Ans: 

Intervals of the given function are$\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\frac{\mathbf{-}\mathbf{1}}{\mathbf{3}}\mathbf{]}\mathbf{\cup }\mathbf{[}\frac{\mathbf{-}\mathbf{1}}{\mathbf{3}}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Always Decreasing function.

Step 1. Given information:

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{ }\frac{\mathbf{3}\mathbf{x}\mathbf{+}\mathbf{1}}{{\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{1}}$

Step 2. Finding the derivative of the function:

$$\begin{gathered} f\left(x\right)=\frac{3x+1}{x^2-1} \\ {f}^{\text{'}}\left(x\right)=\frac{(x^2-1)\left(3\right)-(3x+1)\left(2x\right)}{{\left(x^2-1\right)}^2} \frac{ \partial }{\partial x}\frac{u}{v}=\frac{v\frac{ \partial }{\partial x}\left(u\right)-u\frac{ \partial }{\partial x}\left(v\right)}{v^2} \\ {f}^{\text{'}}\left(x\right)=\frac{3x^2-3-6x^2-2x}{{\left(x^2-1\right)}^2}=\frac{-3x^2-2x-3}{{\left(x^2-1\right)}^2} \\ let f^{\text{'}}\left(x\right)=0 \\ \therefore \frac{-3x^2-2x-3}{{\left(x^2-1\right)}^2}=0 \\ \Rightarrow -3x^2-2x-3={\left(x^2-1\right)}^2 \\ \Rightarrow -3x^2-2x-3=x^4-2x^2+1 \\ \Rightarrow x^4+x^2+2x+2 \\ x=\frac{-1}{3} \end{gathered}$$

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 are $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\frac{\mathbf{-}\mathbf{1}}{\mathbf{3}}\mathbf{]}\mathbf{\cup }\mathbf{[}\frac{\mathbf{-}\mathbf{1}}{\mathbf{3}}\mathbf{,}\mathbf{\infty }\mathbf{)}$

Always Decreasing function.

Step 4. Verifying algebraic answers with graphs :