Find the Critical Points sin(2x)

Problem

ƒ(x)=sin(2*x)

Solution

1. Identify the function ƒ(x)=sin(2*x) and recall that critical points occur where the first derivative ƒ(x)^′ is equal to zero or is undefined.

2. Differentiate the function using the chain rule.

d(sin(2*x))/d(x)=cos(2*x)⋅2

ƒ(x)^′=2*cos(2*x)

3. Set the derivative equal to zero to find the critical values.

2*cos(2*x)=0

4. Solve for x by dividing by 2 and finding where the cosine function is zero.

cos(2*x)=0

5. Apply the general solution for the cosine function being zero, which occurs at odd multiples of π/2

2*x=π/2+n*π

6. Isolate x by dividing the entire expression by 2.

x=π/4+(n*π)/2

Final Answer

x=π/4+(n*π)/2* for any integer *n

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