Find Where Increasing/Decreasing Using Derivatives f(x)=x^2-2x+1

Problem

ƒ(x)=x2−2*x+1

Solution

1. Find the derivative of the function to determine the rate of change.

d(ƒ(x))/d(x)=2*x−2

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

2*x−2=0

2*x=2

x=1

3. Test intervals created by the critical point x=1 to determine the sign of ƒ(x)′
   For x<1 test x=0

ƒ(0)′=2*(0)−2=−2

Since ƒ(x)′<0 the function is decreasing on (−∞,1)

4. Test the interval for x>1
   For x>1 test x=2

ƒ(2)′=2*(2)−2=2

Since ƒ(x)′>0 the function is increasing on (1,∞)

Final Answer

Increasing: *(1,∞), Decreasing: *(−∞,1)

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