Find Where Increasing/Decreasing Using Derivatives g(x)=x^2-2x-80

Problem

g(x)=x2−2*x−80

Solution

1. Find the derivative of the function g(x) using the power rule to determine the slope of the tangent line.

d(g(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 g(x)′ The intervals are (−∞,1) and (1,∞)

4. Evaluate the sign in the first interval (−∞,1) using a test point such as x=0

g(0)′=2*(0)−2=−2

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

5. Evaluate the sign in the second interval (1,∞) using a test point such as x=2

g(2)′=2*(2)−2=2

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

Final Answer

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

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