Find the Local Maxima and Minima f(x)=e^(1-20x+5x^2)

Problem

ƒ(x)=e(1−20*x+5*x2)

Solution

1. Find the first derivative of the function using the chain rule, where d(eu)/d(x)=eud(u)/d(x)

d(ƒ(x))/d(x)=e(1−20*x+5*x2)⋅(−20+10*x)

2. Set the derivative to zero to find the critical points.

e(1−20*x+5*x2)⋅(10*x−20)=0

3. Solve for x by noting that the exponential term e(1−20*x+5*x2) is always positive and never zero.

10*x−20=0

10*x=20

x=2

4. Find the second derivative using the product rule to determine the nature of the critical point.

d2(ƒ(x))/(d(x)2)=e(1−20*x+5*x2)⋅(10)+(10*x−20)⋅e(1−20*x+5*x2)⋅(10*x−20)

5. Evaluate the second derivative at x=2

d2(ƒ(2))/(d(x)2)=e(1−20*(2)+5*(2))⋅10+(10*(2)−20)2⋅e(1−20*(2)+5*(2))

d2(ƒ(2))/(d(x)2)=10*e(−19)+0

d2(ƒ(2))/(d(x)2)=10*e(−19)

6. Apply the second derivative test by observing that 10*e(−19)>0 which indicates a local minimum at x=2

ƒ(2)=e(1−20*(2)+5*(2))

ƒ(2)=e(1−40+20)

ƒ(2)=e(−19)

Final Answer

Local Minimum: *(2,e(−19)), Local Maximum: None

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