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

Problem

ƒ(x)=x*e(−x)

Solution

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

d(ƒ(x))/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(ƒ(x))/d(x)=x*(−e(−x))+e(1)(−x)

d(ƒ(x))/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.

d2(ƒ(x))/(d(x)2)=d()/d(x)*(e(−x)−x*e(−x))

d2(ƒ(x))/(d(x)2)=−e(−x)−(e(−x)−x*e(−x))

d2(ƒ(x))/(d(x)2)=e(−x)*(x−2)

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

ƒ(1)″=e(−1)*(1−2)

ƒ(1)″=−e(−1)

Since *ƒ(1)″<0*, there is a local maximum at *x=1.

6. Calculate the y-value of the local maximum by substituting x=1 into the original function.

ƒ(1)=1⋅e(−1)

ƒ(1)=1/e

Final Answer

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

Want more problems? Check here!