Question

Let y=f(x) be differentiable. Below are four statements. Select every true statement.

A) If f′(x)=0, x is a critical point.
B) The derivative of a constant function is always zero.
C) The Mean Value Theorem states that if f′(c)=0, c is a critical point.
D) The product rule states that (fg)′=f′g+fg′.

Answer 1

The true statements are A) If f′(x)=0, x is a critical point and B) The derivative of a constant function is always zero.

Step-by-step explanation:

The true statements are A) If f′(x)=0, x is a critical point and B) The derivative of a constant function is always zero.

Statement A is true because if the derivative f'(x) is equal to zero at a point x, then x is a critical point of the function. This means that the function may have a local maximum, local minimum, or a point of inflection at x.

Statement B is true because the derivative of a constant function is always zero. This is because a constant function does not change as x varies, so its rate of change (the derivative) is zero.