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

Problem

ƒ(x)=e(1−20*x+5*x2)

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)=eu where u=1−20*x+5*x2

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

d(1−20*x+5*x2)/d(x)=−20+10*x

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

ƒ(x)′=e(1−20*x+5*x2)⋅(−20+10*x)

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

e(1−20*x+5*x2)⋅(−20+10*x)=0

6. Solve for x Since the exponential term e(1−20*x+5*x2) 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

Want more problems? Check here!