Find the Local Maxima and Minima y=xe^(-x)

Problem

y=x*e(−x)

Solution

1. Find the first derivative of the function using the product rule, where (d(u)*v)/d(x)=ud(v)/d(x)+vd(u)/d(x)

d(y)/d(x)=xd(e(−x))/d(x)+e(−x)d(x)/d(x)

2. Apply the chain rule to the exponential term and simplify the expression.

d(y)/d(x)=x*(−e(−x))+e(1)(−x)

d(y)/d(x)=e(−x)*(1−x)

3. Identify critical points by setting the first derivative equal to zero and solving for x

e(−x)*(1−x)=0

1−x=0

x=1

4. Find the second derivative to apply the Second Derivative Test, using the product rule again on e(−x)*(1−x)

d2(y)/(d(x)2)=e(−x)*(−1)+(1−x)*(−e(−x))

d2(y)/(d(x)2)=−e(−x)−e(−x)+x*e(−x)

d2(y)/(d(x)2)=e(−x)*(x−2)

5. Evaluate the second derivative at the critical point x=1 to determine the nature of the point.

d2(y)/(d(x)2)|=e(−1)*(1−2)

d2(y)/(d(x)2)|=−e(−1)

Since −e(−1)<0 a local maximum occurs at x=1

6. Calculate the y-coordinate for the local maximum by substituting x=1 back into the original function.

y=(1)*e(−(1))

y=e(−1)

7. Analyze for local minima by observing the behavior of the derivative. Since there are no other critical points and the function is continuous, there are no local minima.

Final Answer

Local Maximum: *(1,e(−1)), Local Minimum: None

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