Find the Local Maxima and Minima f(x)=x^2-4x+7

Problem

ƒ(x)=x2−4*x+7

Solution

1. Find the first derivative of the function to determine the slope of the tangent line.

d(ƒ(x))/d(x)=2*x−4

2. Set the derivative to zero to find the critical points where the slope is horizontal.

2*x−4=0

3. Solve for x to identify the location of the critical point.

2*x=4

x=2

4. Find the second derivative to apply the Second Derivative Test for concavity.

d2(ƒ(x))/(d(x)2)=2

5. Determine the nature of the point by checking the sign of the second derivative. Since 2>0 the function is concave up at x=2 indicating a local minimum.

ƒ(2)″=2

6. Calculate the y-coordinate by substituting x=2 back into the original function.

ƒ(2)=(2)2−4*(2)+7

ƒ(2)=4−8+7

ƒ(2)=3

Final Answer

Local Minimum: *(2,3), Local Maxima: None

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