Find the Local Maxima and Minima f(x)=x natural log of x

Problem

ƒ(x)=x*ln(x)

Solution

1. Identify the domain of the function. Since the natural logarithm is only defined for positive values, the domain is x>0

2. Find the first derivative using the product rule.

(d(x)*ln(x))/d(x)=ln(x)⋅d(x)/d(x)+x⋅d(ln(x))/d(x)

(d(x)*ln(x))/d(x)=ln(x)+x⋅1/x

ƒ(x)′=ln(x)+1

3. Determine the critical points by setting the first derivative equal to zero.

ln(x)+1=0

ln(x)=−1

x=e(−1)

x=1/e

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

ƒ(x)″=(d(ln(x))+1)/d(x)

ƒ(x)″=1/x

5. Evaluate the second derivative at the critical point x=1/e

ƒ(1/e)″=1/(1/e)

ƒ(1/e)″=e

Since e>0 the function has a local minimum at x=1/e

6. Calculate the y-coordinate of the local minimum.

ƒ(1/e)=1/e*ln(1/e)

ƒ(1/e)=1/e*(−1)

ƒ(1/e)=−1/e

Final Answer

Local Minimum at *(1/e,−1/e), No Local Maxima

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