Find the Local Maxima and Minima f(x)=5-x-x^2

Problem

ƒ(x)=5−x−x2

Solution

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

d(ƒ(x))/d(x)=−1−2*x

2. Set the derivative to zero to solve for the critical value of x

−1−2*x=0

−2*x=1

x=−1/2

3. Find the second derivative to apply the Second Derivative Test and determine the concavity at the critical point.

d2(ƒ(x))/(d(x)2)=−2

4. Evaluate the concavity at x=−1/2 Since the second derivative is −2 which is less than 0 the function is concave down, indicating a local maximum.

ƒ″*(−1/2)=−2<0

5. Calculate the y-coordinate of the local maximum by substituting x=−1/2 back into the original function.

ƒ*(−1/2)=5−(−1/2)−(−1/2)2

ƒ*(−1/2)=5+1/2−1/4

ƒ*(−1/2)=20/4+2/4−1/4=21/4

Final Answer

Local Maximum: *(−1/2,21/4), Local Minimum: None

Want more problems? Check here!