Question
find the minimum value of the function f, of, xf(x).
f(x)=x^2−2x−24
f(x)=x^2−2x−24
Answers
GPT 3.5
To find the minimum value of the function f(x) = x^2 - 2x - 24, we first need to find the derivative of f(x) and set it equal to zero to find the critical points.
f'(x) = 2x - 2
Setting f'(x) equal to zero:
2x - 2 = 0
2x = 2
x = 1
So, the critical point is x = 1.
Next, we need to determine if this critical point is a minimum or maximum. We can do this by using the second derivative test.
Taking the second derivative of f(x):
f''(x) = 2
Since f''(x) is positive for all x, this indicates that x = 1 is a local minimum.
To find the minimum value of the function, plug x = 1 back into the function:
f(1) = 1^2 - 2(1) - 24
f(1) = 1 - 2 - 24
f(1) = -25
Therefore, the minimum value of the function f(x) = x^2 - 2x - 24 is -25.
f'(x) = 2x - 2
Setting f'(x) equal to zero:
2x - 2 = 0
2x = 2
x = 1
So, the critical point is x = 1.
Next, we need to determine if this critical point is a minimum or maximum. We can do this by using the second derivative test.
Taking the second derivative of f(x):
f''(x) = 2
Since f''(x) is positive for all x, this indicates that x = 1 is a local minimum.
To find the minimum value of the function, plug x = 1 back into the function:
f(1) = 1^2 - 2(1) - 24
f(1) = 1 - 2 - 24
f(1) = -25
Therefore, the minimum value of the function f(x) = x^2 - 2x - 24 is -25.