Q. 45 - 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.

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathbf{\pi }\mathbf{\right(}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{\left)}\mathbf{\right)}$

Step-by-Step Solution

Answer

Ans: The local maximum of the function f(x) is at an odd integer.

The local minimum of the function f(x) is at an even integer.

Step 1. Given Information:

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathbf{\left(}\mathbf{\pi }\mathbf{\right(}\mathbf{x}\mathbf{+}\mathbf{1}\mathbf{\left)}\mathbf{\right)}$

Step 2. Finding the derivative of the function:

$$\begin{gathered} f\left(x\right)=\cos \left(\mathrm{\pi }\right(x+1\left)\right) \\ f\text{'}\left(x\right)=-\sin \left(\mathrm{\pi }\right(x+1\left)\right)·\mathrm{\pi } \\ f\text{'}\left(x\right)=-\mathrm{\pi sin}\left(\mathrm{\pi }\right(x+1\left)\right) \\ lets, f\text{'}\left(x\right)=0 \\ \therefore -\mathrm{\pi sin}\left(\mathrm{\pi }\right(x+1\left)\right)=0 \\ \sin \left(\mathrm{\pi }\right(x+1\left)\right)=0 \\ \cancel{ \mathrm{\pi }}(x+1)=k\cancel{\mathrm{\pi }} [\mathrm{k} \mathrm{is} \mathrm{any} \mathrm{integer}, \mathrm{k}=...-2,-1,0,1,2,3,... ] \\ x+1=k \\ x=k-1 \\ the critical points is x=k-1 \end{gathered}$$

Step 3. Substituting the values into the function equation:

$$\begin{gathered} f(k-1)=\cos \left(\mathrm{\pi }\right((\mathrm{k}-1)+1\left)\right) \\ =\cos k\mathrm{\pi } \\ =\left\{\begin{array}{ll}-1;& \mathrm{when} \mathrm{k} \mathrm{is} \mathrm{odd}\\ 1;& \mathrm{when} \mathrm{k} \mathrm{is} \mathrm{even}\end{array}\right. \end{gathered}$$

The local maximum of the function f(x) is at an odd integer.

The local minimum of the function f(x) is at an even integer.

Step 4. Verifying algebraic answers with graphs :