Graph 3x natural log of x

Problem

ƒ(x)=3*x*ln(x)

Solution

1. Determine 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

3*x*ln(x)=0

ln(x)=0⇒x=1

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

3. Determine the limit as x approaches 0 from the right.

(lim_x→0)(3)*x*ln(x)=0

The graph approaches the origin (0,0) but does not include it.

4. Find the derivative using the product rule to identify critical points.

(d(3)*x*ln(x))/d(x)=3*ln(x)+3*x⋅1/x

d(ƒ(x))/d(x)=3*ln(x)+3

5. Locate the local minimum by setting the derivative to zero.

3*ln(x)+3=0

ln(x)=−1⇒x=e(−1)≈0.368

The y-value is ƒ(e(−1))=3*(e(−1))*(−1)=−3*e(−1)≈−1.104

6. Determine concavity by finding the second derivative.

d(3*ln(x)+3)/d(x)=3/x

Since x>0 the second derivative is always positive, meaning the graph is always concave up.

7. Analyze end behavior as x goes to infinity.

(lim_x→∞)(3)*x*ln(x)=∞

The function grows without bound as x increases.

Final Answer

ƒ(x)=3*x*ln(x)

Want more problems? Check here!