Suppose you are given the function f(x)=e3x−2e2x+1 . Which of the following best represents the location of the global minimum of the function? (1 point) Responses x=−0.48 x = − 0.48 x=−0.12 x = − 0.12 x = 0.00 x = 0.00 x = 0.48 x = 0.48 x = 0.29 x = 0.29 Question 2 Part B Which of the following best represents the location of the inflection point of the function? (1 point) Responses x=−0.12 x = − 0.12 x=−0.48 x = − 0.48 x = 0.00 x = 0.00 x = 0.29 x = 0.29 x = 0.48 x = 0.48 Skip to navigation
To find the location of the global minimum, we need to find the critical points of the function f(x) and determine which one gives the minimum value.
First, let's find the derivative of f(x):
f'(x) = 3e^3x - 4e^2x
To find the critical points, we set the derivative equal to zero and solve for x:
3e^3x - 4e^2x = 0
e^3x = (4/3)e^2x
Taking the natural logarithm of both sides:
3x = ln(4/3) + 2x
x = ln(4/3)
From the given responses, x = 0.29 is the closest value to ln(4/3), so the location of the global minimum is x = 0.29.
To find the inflection point, we need to find the second derivative of f(x). Taking the derivative of f'(x):
f''(x) = 9e^3x - 8e^2x
To find the inflection point, we need to find the x-value where the second derivative equals zero. Setting f''(x) = 0:
9e^3x - 8e^2x = 0
e^3x = (8/9)e^2x
Taking the natural logarithm of both sides:
3x = ln(8/9) + 2x
x = ln(8/9)
From the given responses, x = -0.12 is the closest value to ln(8/9), so the location of the inflection point is x = -0.12.
Therefore, the correct answers are:
1. The location of the global minimum is x = 0.29.
2. The location of the inflection point is x = -0.12.