Find the Critical Points sin(x)-cos(x)

Problem

ƒ(x)=sin(x)−cos(x)

Solution

1. Find the derivative of the function ƒ(x) with respect to x to determine the rate of change.

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

d(ƒ(x))/d(x)=cos(x)+sin(x)

2. Set the derivative to zero to find the values of x where the slope of the tangent line is horizontal.

cos(x)+sin(x)=0

3. Rearrange the equation to isolate the trigonometric functions.

sin(x)=−cos(x)

4. Divide by cos(x) (assuming cos(x)≠0 to express the equation in terms of the tangent function.

sin(x)/cos(x)=−1

tan(x)=−1

5. Solve for x by finding the angles where the tangent is −1 These occur in the second and fourth quadrants.

x=(3*π)/4+n*π

Final Answer

Critical Points: *x=(3*π)/4+n*π* for any integer *n

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