Find the Critical Points xe^x

Problem

x*ex

Solution

1. Identify the function ƒ(x)=x*ex 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)*ex)/d(x)=xd(ex)/d(x)+exd(x)/d(x)

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

ƒ(x)′=x*ex+ex

5. Factor out the common term ex

ƒ(x)′=ex*(x+1)

6. Set the derivative to zero to solve for x

ex*(x+1)=0

7. Solve for x by analyzing the factors. Since ex 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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