Find the Local Maxima and Minima f(x)=x^2-8 natural log of x

Problem

ƒ(x)=x2−8*ln(x)

Solution

1. Determine the domain of the function. Since the natural logarithm ln(x) is only defined for positive values, the domain is x>0

2. Find the first derivative of ƒ(x) with respect to x

d(ƒ(x))/d(x)=2*x−8/x

3. Identify critical points by setting the first derivative equal to zero and solving for x within the domain.

2*x−8/x=0

2*x=8/x

2*x2=8

x2=4

x=2

Note: x=−2 is excluded because it is outside the domain x>0

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

d2(ƒ(x))/(d(x)2)=2+8/(x2)

5. Evaluate the second derivative at the critical point x=2

ƒ(2)″=2+8/2

ƒ(2)″=2+2=4

Since ƒ(2)″>0 the function has a local minimum at x=2

6. Calculate the function value at the local minimum.

ƒ(2)=2−8*ln(2)

ƒ(2)=4−8*ln(2)

Final Answer

Local Minimum: *(2,4−8*ln(2))

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