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

Problem

ƒ(x)=ln(x)/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 quotient rule, where u=ln(x) and v=x

d(ƒ(x))/d(x)=(x⋅d(ln(x))/d(x)−ln(x)⋅d(x)/d(x))/(x2)

3. Simplify the derivative by substituting the derivatives of the individual terms.

d(ƒ(x))/d(x)=(x⋅1/x−ln(x)⋅1)/(x2)

d(ƒ(x))/d(x)=(1−ln(x))/(x2)

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

(1−ln(x))/(x2)=0

1−ln(x)=0

ln(x)=1

x=e

5. Apply the first derivative test to determine the nature of the critical point. For 0<x<e d(ƒ(x))/d(x)>0 (the function is increasing). For x>e d(ƒ(x))/d(x)<0 (the function is decreasing).

6. Calculate the function value at the critical point x=e

ƒ(e)=ln(e)/e

ƒ(e)=1/e

7. Conclude that since the function changes from increasing to decreasing at x=e there is a local maximum at that point and no local minimum.

Final Answer

Local Maxima: *(e,1/e), Local Minima: None

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