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

Problem

ƒ(x)=x+cos(x)

Solution

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

d(ƒ(x))/d(x)=1−sin(x)

2. Set the derivative to zero to find the critical values of x

1−sin(x)=0

sin(x)=1

3. Solve for x within the general domain.

x=π/2+2*n*π

where n is any integer.

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

d2(ƒ(x))/(d(x)2)=−cos(x)

5. Evaluate the second derivative at the critical points x=π/2+2*n*π

ƒ″*(π/2+2*n*π)=−cos(π/2+2*n*π)

ƒ″*(π/2+2*n*π)=0

6. Analyze the result since the Second Derivative Test is inconclusive. Observe the first derivative ƒ(x)′=1−sin(x) Since sin(x)≤1 for all x then 1−sin(x)≥0 The derivative never changes sign; it is always non-negative.

7. Conclude that because the function is monotonically increasing and the derivative does not change sign at the critical points, these points are horizontal points of inflection rather than local extrema.

Final Answer

ƒ(x)=x+cos(x)* has no local maxima or minima

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