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

Problem

ƒ(x)=e^(1−20*x+5*x^2)

Solution

1. Identify the definition of critical points, which occur where the first derivative ƒ(x)^′ is equal to zero or is undefined.

2. Apply the chain rule to find the derivative of the function ƒ(x)=e^u where u=1−20*x+5*x^2

3. Calculate the derivative of the exponent u with respect to x

d(1−20*x+5*x^2)/d(x)=−20+10*x

4. Combine the results to form the full derivative ƒ(x)^′

ƒ(x)^′=e^(1−20*x+5*x^2)⋅(−20+10*x)

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

e^(1−20*x+5*x^2)⋅(−20+10*x)=0

6. Solve for x Since the exponential term e^(1−20*x+5*x^2) is always positive and never zero, we only need to solve the linear factor.

−20+10*x=0

7. Isolate the variable x

10*x=20

x=2

Final Answer

x=2

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