Graph x/( natural log of x)

Problem

ƒ(x)=x/ln(x)

Solution

1. Identify the domain of the function. Since the natural logarithm ln(x) is defined for x>0 and the denominator cannot be zero (ln(x)≠0⇒x≠1, the domain is (0,1)∪(1,∞)

2. Find the vertical asymptote by checking the limit as x approaches 1 Since ln(1)=0 there is a vertical asymptote at x=1

3. Determine the end behavior by calculating limits. As x→0 ln(x)→−∞ so ƒ(x)→0 As x→∞ applying L'Hôpital's Rule shows ƒ(x)→∞

4. Find the first derivative to locate critical points using the quotient rule.

d(ƒ(x))/d(x)=(ln(x)−1)/((ln(x))2)

5. Solve for critical points by setting the numerator to zero.

ln(x)−1=0⇒x=e

6. Analyze intervals of increase and decrease. The function decreases on (0,1) and (1,e) and increases on (e,∞) There is a local minimum at (e,e)

7. Find the second derivative to determine concavity.

d2(ƒ(x))/(d(x)2)=(2−ln(x))/(x*(ln(x))3)

8. Identify concavity and inflection points. The function is concave down on (0,1) concave up on (1,e2) and concave down on (e2,∞) with an inflection point at x=e2

Final Answer

ƒ(x)=x/ln(x)

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