Graph x natural log of x-x

Problem

ƒ(x)=x*ln(x)−x

Solution

1. Identify 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 x-intercept by setting ƒ(x)=0

x*ln(x)−x=0

x*(ln(x)−1)=0

Since x>0 we solve ln(x)−1=0 which gives ln(x)=1 so x=e≈2.718

3. Find the derivative to determine the slope and critical points.

(d(x)*ln(x)−x)/d(x)=ln(x)+x⋅1/x−1

d(ƒ(x))/d(x)=ln(x)

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

ln(x)=0

x=e0=1

5. Find the y-coordinate of the critical point.

ƒ(1)=1*ln(1)−1

ƒ(1)=−1

Since d(ƒ(x))/d(x)<0 for 0<x<1 and d(ƒ(x))/d(x)>0 for x>1 the point (1,−1) is a local minimum.

6. Analyze the limit as x approaches zero from the right.

(lim_x→0)(x*ln(x)−x)=0

The graph starts near the origin (0,0) but is not defined at x=0

7. Find the second derivative to determine concavity.

d(ln(x))/d(x)=1/x

Since 1/x>0 for all x in the domain, the graph is always concave up.

Final Answer

ƒ(x)=x*ln(x)−x

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