Find the Critical Points e^(1-20x+5x^2)

Problem

e(1−20*x+5*x2)

Solution

1. Identify the function as ƒ(x)=e(1−20*x+5*x2) Critical points occur where the first derivative ƒ(x)′ is equal to zero or is undefined.

2. Apply the chain rule to find the derivative. The derivative of eu is eu⋅d(u)/d(x)

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

3. Differentiate the exponent using the power rule.

d(1−20*x+5*x2)/d(x)=−20+10*x

4. Combine the results to express the full derivative.

ƒ(x)′=(−20+10*x)*e(1−20*x+5*x2)

5. Set the derivative to zero to solve for x Since the exponential term e(1−20*x+5*x2) is always positive and never zero, we only need to solve for the linear factor.

−20+10*x=0

6. Solve for x by adding 20 to both sides and dividing by 10.

10*x=20

x=2

Final Answer

x=2

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