Graph ( natural log of x)^2

Problem

(ln(x))2

Solution

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

2. Find the x-intercept by setting the function to zero.

(ln(x))2=0

ln(x)=0

x=e0=1

The graph touches the x-axis at (1,0)

3. Determine the behavior as x approaches the boundaries of the domain. As x→0 ln(x)→−∞ so (ln(x))2→∞ This indicates a vertical asymptote at x=0 As x→∞ ln(x)→∞ so (ln(x))2→∞

4. Find the first derivative to determine intervals of increase and decrease using the chain rule.

d(ln(x))/d(x)=2*ln(x)⋅1/x

d(ln(x))/d(x)=(2*ln(x))/x

5. Identify critical points by setting the derivative to zero.

(2*ln(x))/x=0

ln(x)=0

x=1

For 0<x<1 the derivative is negative (decreasing). For x>1 the derivative is positive (increasing). There is a local minimum at (1,0)

6. Find the second derivative to determine concavity and inflection points using the quotient rule.

d()/d(x)(2*ln(x))/x=(x⋅2/x−2*ln(x)⋅1)/(x2)

d2(ln(x))/(d(x)2)=(2−2*ln(x))/(x2)

7. Identify inflection points by setting the second derivative to zero.

2−2*ln(x)=0

ln(x)=1

x=e

The graph is concave up on (0,e) and concave down on (e,∞) The inflection point is (e,1)

Final Answer

y=(ln(x))2

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