Question
find the minimum or maximum of the function below
f(x)=4x^2+2
f(x)=4x^2+2
Answers
GPT 3.5
To find the minimum or maximum of a function, we need to first take the derivative of the function and set it equal to zero to find the critical points.
Given function: f(x) = 4x^2 + 2
Taking the derivative of f(x) with respect to x, we get:
f'(x) = 8x
Setting the derivative equal to zero to find critical points:
8x = 0
x = 0
Now, we need to determine if this critical point is a minimum or maximum. To do this, we can use the second derivative test.
Taking the second derivative of f(x) to analyze concavity:
f''(x) = 8
Since the second derivative is positive (8), the critical point x = 0 corresponds to a minimum.
Therefore, the minimum value of the function f(x) = 4x^2 + 2 is at x = 0, and the minimum value is f(0) = 2.
Given function: f(x) = 4x^2 + 2
Taking the derivative of f(x) with respect to x, we get:
f'(x) = 8x
Setting the derivative equal to zero to find critical points:
8x = 0
x = 0
Now, we need to determine if this critical point is a minimum or maximum. To do this, we can use the second derivative test.
Taking the second derivative of f(x) to analyze concavity:
f''(x) = 8
Since the second derivative is positive (8), the critical point x = 0 corresponds to a minimum.
Therefore, the minimum value of the function f(x) = 4x^2 + 2 is at x = 0, and the minimum value is f(0) = 2.