Graph e^x natural log of x

Problem

ex*ln(x)

Solution

1. Identify the domain of the function ƒ(x)=ex*ln(x) Since the natural logarithm ln(x) is only defined for positive values, the domain is x>0

2. Determine the x-intercept by setting ƒ(x)=0 Since ex is never zero, we solve ln(x)=0 which gives x=1 The intercept is at (1,0)

3. Analyze the end behavior as x→0 As x approaches 0 from the right, ex→1 and ln(x)→−∞ Therefore, (lim_x→0)(ex)*ln(x)=−∞ indicating a vertical asymptote at x=0

4. Analyze the end behavior as x→∞ Both ex and ln(x) grow without bound as x increases, so (lim_x→∞)(ex)*ln(x)=∞

5. Find the derivative to determine the slope and critical points using the product rule.

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

6. Evaluate the derivative for x>0 Since ex is always positive, and for x≥1 both ln(x) and 1/x are non-negative, the function is strictly increasing for x≥1 For 0<x<1 the function increases from −∞ until it crosses the x-axis at x=1 and continues upward.

7. Sketch the graph starting from the vertical asymptote at the y-axis, passing through (1,0) and curving upward steeply toward infinity.

Final Answer

y=ex*ln(x)

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