the local linear approximation of a function f will always be greater than or equal to the function's value if, for all x in an interval containing the point of tangency,
f'(x) <0
f'(x)>0
f"(x) <0
f"(x) >0
the line must lie above the curve, so the graph must be concave down.
So, what do you think?
f"(x) < 0 ???????
right
but I posted this answer in my homework and the teacher marked (zero). I don't know what happened
I can not imagine what happened but agree that the second derivative is negative.
To determine the conditions where the local linear approximation of function f will always be greater than or equal to the function's value, we need to understand the relationship between the slope of the function's tangent line and the concavity of the function.
The local linear approximation of a function f at a point (a, f(a)) is given by the equation of the tangent line at that point, which can be approximated by:
L(x) = f(a) + f'(a)(x - a),
where f'(a) represents the derivative of f evaluated at a.
Now, let's analyze the given options:
1. f'(x) < 0:
If the derivative of f is negative in an interval containing the point of tangency, it means that the function is decreasing around that point. In this scenario, the local linear approximation would be a line with a negative slope. Therefore, the approximation will be less than the function's value.
2. f'(x) > 0:
If the derivative of f is positive in an interval containing the point of tangency, it means that the function is increasing around that point. In this case, the local linear approximation would be a line with a positive slope. Thus, the approximation will be greater than the function's value.
3. f"(x) < 0:
This option refers to the concavity of the function. If the second derivative of f is negative in an interval, it means that the function is concave down around the point of tangency. However, the concavity does not necessarily determine whether the local linear approximation is greater or smaller than the function's value. We need to consider the slope of the tangent line as well.
4. f"(x) > 0:
Similar to the previous case, the positive second derivative indicates that the function is concave up around the point of tangency. Yet, it does not provide sufficient information about the local linear approximation's relationship with the function's value. The slope of the tangent line is crucial to determine the approximation's behavior.
Therefore, based on the given options, the correct condition for the local linear approximation to be greater than or equal to the function's value is when f'(x) > 0, which implies that the function is increasing around the point of tangency.