1. Under what conditions is the tangent to f(x) to the point (a,f(a)) horizontal?

A.f'(a) > 0
B.f'(a)=0
C.f'(a) < 0
D. f'(a) is undefined

The derivative is zero where the function has a horizontal tangent.

f ' (a) = 0

Well, let's see. If the tangent line to f(x) at the point (a, f(a)) is horizontal, that means the slope of the tangent line is zero. In other words, the derivative of f(x) at the point a, which is f'(a), must be zero. So, the correct answer is B. f'(a) = 0. Why did the function go to the beach? To get a tan-gent!

The condition for the tangent to f(x) to be horizontal at the point (a,f(a)) can be determined by analyzing the derivative of the function, f'(a).

When the derivative of f(x) is positive (f'(a) > 0), the tangent line will have a positive slope and will be upward sloping. Therefore, it is not horizontal.

When the derivative of f(x) is equal to zero (f'(a) = 0), the tangent line will have a slope of 0, which means it will be horizontal. So option B, f'(a) = 0, is the correct answer.

When the derivative of f(x) is negative (f'(a) < 0), the tangent line will have a negative slope and will be downward sloping. Therefore, it is not horizontal.

When the derivative of f(x) is undefined, it means that the function is not differentiable at that point. This could happen, for example, when there is a sharp corner or vertical tangent at that point. So option D, f'(a) is undefined, does not necessarily mean that the tangent is horizontal.

Therefore, the correct answer is option B, f'(a) = 0.

To determine the conditions under which the tangent to the function f(x) at the point (a, f(a)) is horizontal, we need to analyze the behavior of the derivative of the function at that point.

The derivative of a function represents the rate of change of the function at a particular point. If the derivative is positive, it indicates that the function is increasing; if it is negative, it indicates the function is decreasing. A derivative equal to zero indicates that the function has a local extremum (maximum or minimum) at that point.

In this case, we are looking for a horizontal tangent, which means that the slope of the tangent line (represented by the derivative) is zero. Hence, the correct option is B. f'(a) = 0.

To find this condition algebraically, you would need to calculate the derivative of the function f(x) and then evaluate it at the point (a, f(a)). If the resulting value is zero, it indicates that the tangent at that point is horizontal.