The function's value will always be greater than or equal to the local linear approximation of a function f if, for all x in an interval containing the point of tangency,
f " > 0
f " < 0
f ' > 0
f ' < 0
see other post.
To determine if the function's value will always be greater than or equal to the local linear approximation of a function f, we need to consider the concavity of the function.
The local linear approximation is given by the equation y = f(a) + f'(a)(x - a), where a is the point of tangency. This linear equation represents the tangent line to the function at the point a.
Now, let's examine the options:
- f " > 0: This condition implies that the second derivative of the function f is positive. If the second derivative is positive, it means that the function is concave up. In this case, the function's value will always be greater than or equal to the local linear approximation.
- f " < 0: This condition implies that the second derivative of the function f is negative. If the second derivative is negative, it means that the function is concave down. In this case, the function's value may be less than the local linear approximation.
- f ' > 0: This condition implies that the first derivative of the function f is positive. If the first derivative is positive, it means that the function is increasing. However, it does not give enough information about the concavity of the function. Therefore, we cannot conclude if the function's value will always be greater than or equal to the local linear approximation.
- f ' < 0: This condition implies that the first derivative of the function f is negative. If the first derivative is negative, it means that the function is decreasing. Similar to the previous option, it does not provide enough information about the concavity of the function.
Therefore, the correct answer is f " > 0, meaning that the second derivative of the function f is positive. This condition ensures that the function is concave up and guarantees that the function's value will always be greater than or equal to the local linear approximation.