What does it mean if the second derivative is concave up?

Taking the second derivative actually tells us if the slope continually increases or decreases. When the second derivative is positive, the function is concave upward. When the second derivative is negative, the function is concave downward.

Is the second derivative the slope of the first derivative?

The second derivative is acceleration or how fast velocity changes. Graphically, the first derivative gives the slope of the graph at a point. The second derivative tells whether the curve is concave up or concave down at that point.

What is the relationship between the inflection point and the first and second derivative?

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 happens when second derivative is undefined?

The concavity changes “at” x=0 . But, since f(0) is undefined, there is no inflection point for the graph of this function. f(x)=3√x is concave up for x<0 and concave down for x>0 . The second derivative is undefined at x=0 .

What does the first derivative of a curve tell you?

The first derivative of a function is an expression which tells us the slope of a tangent line to the curve at any instant. Because of this definition, the first derivative of a function tells us much about the function. If is positive, then must be increasing. If is negative, then must be decreasing.

When a function is concave up and concave down?

When the function y = f (x) is concave up, the graph of its derivative y = f ‘(x) is increasing. When the function y = f (x) is concave down, the graph of its derivative y = f ‘(x) is decreasing.

How does second derivative determine concavity?

The 2nd derivative is tells you how the slope of the tangent line to the graph is changing. If you’re moving from left to right, and the slope of the tangent line is increasing and the so the 2nd derivative is postitive, then the tangent line is rotating counter-clockwise. That makes the graph concave up.

What is second derivative rule?

If the second derivative is positive over an interval, indicating that the change of the slope of the tangent line is increasing, the graph is concave up over that interval. CONCAVITY TEST: If f ”(x) < 0 over an interval, then the graph of f is concave upward over this interval.

How do you find second derivative of inflection points?

An inflection point is a point on the graph of a function at which the concavity changes. Points of inflection can occur where the second derivative is zero. In other words, solve f ” = 0 to find the potential inflection points. Even if f ”(c) = 0, you can’t conclude that there is an inflection at x = c.