Find the Local Maxima and Minima f(x)=cos(pix)

Problem

ƒ(x)=cos(π*x)

Solution

1. Find the first derivative of the function with respect to x using the chain rule.

d(cos(π*x))/d(x)=−π*sin(π*x)

2. Identify critical points by setting the first derivative equal to zero and solving for x

−π*sin(π*x)=0

sin(π*x)=0

π*x=n*π

x=n

where n is any integer.

3. Find the second derivative to apply the second derivative test for local extrema.

(d(−)*π*sin(π*x))/d(x)=−π2*cos(π*x)

4. Evaluate the second derivative at the critical points x=n

ƒ(n)″=−π2*cos(n*π)

ƒ(n)″=−π2*(−1)n

5. Determine the nature of extrema based on the sign of the second derivative.
   If n is even (n=2*k, then ƒ″*(2*k)=−π2*(1)=−π2<0 indicating a local maximum.
   If n is odd (n=2*k+1, then ƒ″*(2*k+1)=−π2*(−1)=π2>0 indicating a local minimum.

6. Calculate the function values at these points.
   For even n ƒ(n)=cos(2*k*π)=1
   For odd n ƒ(n)=cos((2*k+1)*π)=−1

Final Answer

Local Maxima: *(n,1)* for even *n; Local Minima: *(n,−1)* for odd *n

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