Find the Critical Points xe^x

Problem

x*e^x

Solution

1. Identify the function ƒ(x)=x*e^x and the condition for critical points, which occurs where the first derivative ƒ(x)^′=0 or is undefined.

2. Apply the product rule to find the derivative, where (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

3. Differentiate the expression.

(d(x)*e^x)/d(x)=xd(e^x)/d(x)+e^xd(x)/d(x)

4. Simplify the derivative by evaluating the basic derivatives d(e^x)/d(x)=e^x and d(x)/d(x)=1

ƒ(x)^′=x*e^x+e^x

5. Factor out the common term e^x

ƒ(x)^′=e^x*(x+1)

6. Set the derivative to zero to solve for x

e^x*(x+1)=0

7. Solve for x by analyzing the factors. Since e^x is never zero for any real x we set the other factor to zero.

x+1=0

x=−1

Final Answer

x=−1

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