What does it mean when the first and second derivative equals zero?

Inflection points are where the function changes concavity. Since concave up corresponds to a positive second derivative and concave down corresponds to a negative second derivative, then when the function changes from concave up to concave down (or vise versa) the second derivative must equal zero at that point.

What does the first second and third derivative tell us?

A first derivative expresses our rate of change (like an increase in distance: ). A second derivative expresses our rate of change of our rate of change (like an increase in velocity: ). A third derivative expresses a rate of change of our rate of change of our rate of change (like an increase in acceleration).

What does it mean if the second derivative is less than 0?

concave down
The second derivative of f(x) tells us the rate of change of the derivative f (x) of f(x). The second derivative is negative (f (x) < 0): When the second derivative is negative, the function f(x) is concave down.

What does the mean value theorem state?

The Mean Value Theorem states that if a function f is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in the interval (a,b) such that f'(c) is equal to the function’s average rate of change over [a,b].

What does it mean when the derivative is zero?

The derivative of a function, f(x) being zero at a point, p means that p is a stationary point. That is, not “moving” (rate of change is 0). There are a few things that could happen. Either the function has a local maximum, minimum, or saddle point.

What does it mean if the derivative is equal to zero?

The first derivative of a point is the slope of the tangent line at that point. When the slope of the tangent line is 0, the point is either a local minimum or a local maximum. Thus when the first derivative of a point is 0, the point is the location of a local minimum or maximum.

What does 3rd derivative tell you?

The third derivative is the rate of change of the rate of change of the slope. When it is zero, the second derivative is constant, and the rate of the slope changing is fixed. When the third derivative is not zero, then the rate of change of the slope is not constant.

What does negative 2nd derivative mean?

Likewise, if the second derivative is negative, then the first derivative is decreasing, so that. the slope of the tangent line to the function is decreasing as x increases. Graphically, we see this as the curve. of the graph being concave down, that is, shaped like a parabola open downward.

What is first mean value theorem?

This theorem (also known as First Mean Value Theorem) allows to express the increment of a function on an interval through the value of the derivative at an intermediate point of the segment.