Q. 42 - Calculus

Question

Use the first derivative test to determine the local extrema of each function in Exercises 39- 50. Then verify your algebraic answers with graphs from a calculator or graphing utility.

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

Step-by-Step Solution

Answer

Ans: The local minimum of the function f(x) is at $x=1$

The local maximum of the function f(x) is at $x=-5$

Step 1. Given Information:

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

Step 2. Finding the derivative of the function

Rewriting the function by simplifying it,

$$\begin{gathered} f\left(x\right)=\frac{{(x-1)}^2}{x+2} \\ f\text{'}\left(x\right)=\frac{(x+2)2(x-1)-1-(x-1{)}^2-1}{(x+2{)}^2} \\ f\text{'}\left(x\right)=\frac{(x+2)(2x-2)-\left(x^2-2x+1\right)}{(x+2{)}^2} \\ f\text{'}\left(x\right)=\frac{2x^2-2x+4x-4-x^2+2x-1}{(x+2{)}^2} \\ f\text{'}\left(x\right)=\frac{x^2+4x-5}{(x+2{)}^2} \\ let, f\text{'}\left(x\right)=0 \\ \therefore \frac{x^2+4x-5}{(x+2{)}^2}=0 \\ \begin{array}{r}{x}^2+4x-5=0\\ x^2+5x-x-5=0\\ x(x+5)-(x+5)=0\\ (x+5)(x-1)=0\end{array} \\ so, the citical points are x=-5,1 \end{gathered}$$

Step 3. Substituting the values into the function equation:

$$\begin{gathered} f(-5)=\frac{{(-5-1)}^2}{-5+2} \\ =\frac{{(-6)}^2}{-3}=\frac{36}{-3} \\ =-12<0 \\ \mathit{f}\mathbf{\left(}\mathbf{1}\mathbf{\right)}\mathbf{=}\frac{{\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{1}\mathbf{)}}^{\mathbf{2}}}{\mathbf{1}\mathbf{+}\mathbf{2}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\frac{{\mathbf{\left(}\mathbf{0}\mathbf{\right)}}^{\mathbf{2}}}{\mathbf{3}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\mathbf{0}\mathbf{=}\mathbf{0} \\ \mathbf{\therefore }\mathbf{ }\mathit{t}\mathit{h}\mathit{e}\mathbf{ }\mathit{l}\mathit{o}\mathit{c}\mathit{a}\mathit{l}\mathbf{ }\mathit{m}\mathit{i}\mathit{n}\mathit{i}\mathit{m}\mathit{u}\mathit{m}\mathbf{ }\mathit{a}\mathit{t}\mathbf{ }\mathit{x}\mathbf{=}\mathbf{1}\mathbf{:}\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathit{l}\mathit{o}\mathit{c}\mathit{a}\mathit{l}\mathbf{ }\mathit{m}\mathit{a}\mathit{x}\mathit{i}\mathit{m}\mathit{u}\mathit{m}\mathbf{ }\mathit{a}\mathit{t}\mathbf{ }\mathit{x}\mathbf{=}\mathbf{-}\mathbf{5}\mathbf{ } \end{gathered}$$

Step 4. Verifying algebraic answers with graphs :