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

Problem

ƒ(x)=sin(x)+cos(x)

Solution

1. Find the first derivative of the function to identify critical points.

(d(sin(x))+cos(x))/d(x)=cos(x)−sin(x)

2. Set the derivative to zero to solve for x

cos(x)−sin(x)=0

sin(x)=cos(x)

tan(x)=1

3. Identify critical points within the standard interval [0,2*π]

x=π/4,(5*π)/4

4. Find the second derivative to apply the Second Derivative Test.

(d(cos(x))−sin(x))/d(x)=−sin(x)−cos(x)

5. Test the critical point x=π/4 in the second derivative.

ƒ(π/4)″=−sin(π/4)−cos(π/4)

ƒ(π/4)″=−√(,2)/2−√(,2)/2=−√(,2)

Since ƒ(π/4)″<0 a local maximum occurs at x=π/4

6. Test the critical point x=(5*π)/4 in the second derivative.

ƒ((5*π)/4)″=−sin((5*π)/4)−cos((5*π)/4)

ƒ((5*π)/4)″=−(−√(,2)/2)−(−√(,2)/2)=√(,2)

Since ƒ((5*π)/4)″>0 a local minimum occurs at x=(5*π)/4

7. Calculate the function values at these points.

ƒ(π/4)=sin(π/4)+cos(π/4)=√(,2)

ƒ((5*π)/4)=sin((5*π)/4)+cos((5*π)/4)=−√(,2)

Final Answer

Local Maxima: *(π/4+2*n*π,√(,2)), Local Minima: *((5*π)/4+2*n*π,−√(,2))

Want more problems? Check here!