Question

Help with these 2 please?

Answer 1

I and III only

Explanation:

Statement 1 asks "if the derivative of the function is greater than 0 for an interval, will the function increase on that interval?"

The answer is yes.

A derivative is a rate of change. If the rate of change for the entire interval is positive, then that means that the function is increasing, by definition.

Statement 2 asks "if two points have the same value, must there be some point in between them where the slope is zero (horizontal)?"

The answer is not necessarily.

This is the definition of a special version of the Mean Value Theorem called Rolle's Theorem. However, there is one thing missing from the definition to make it correct.

Rolle's Theorem guarantees that if two points are at the same height on a graph, then at some point between those two points, the slope will be 0. However, for this to be true, the function must also be continuous.

If the function is not continuous, then it could have an asymptote between points `a` and `b`. This means that the function could potentially increase to infinity, jump back down to negative infinity, and increase again back up to the starting value (in other words, there would be no point where the slope is zero).

Because the problem doesn't specify that the function is continuous, Statement 2 is false.

Statement 3 asks "If a function is differentiable at a point, is it also continuous at that point?"

The answer is yes.

This is because continuity is in the definition of a derivative.

If the function is not continuous at a given point, then either the point itself doesn't exist (so there's no way it could have a slope at all) or the points around it are not connected to it.

Either way, it would be impossible to connect the point to the points around it to find the "slope" of the tangent at that point.

So the correct choice is "I and III only"