What does it mean when both 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 it mean when the second derivative is equal to zero?

inflection point
The second derivative is zero (f (x) = 0): When the second derivative is zero, it corresponds to a possible inflection point. If the second derivative changes sign around the zero (from positive to negative, or negative to positive), then the point is an inflection point.

What does the second derivative tell you in a word problem?

The sign of the second derivative tells us whether the slope of the tangent line to f is increasing or decreasing. In other words, the second derivative tells us the rate of change of the rate of change of the original function.

What does it mean when the second derivative test fails?

If f (x0) = 0, the test fails and one has to investigate further, by taking more derivatives, or getting more information about the graph. Besides being a maximum or minimum, such a point could also be a horizontal point of inflection.

What if the second derivative test is 0 multivariable?

If D(a, b) = 0 then the second derivative test is inconclusive, and the point (a, b) could be any of a minimum, maximum or saddle point.

What is the meaning of double derivative?

Double differentiation is just rate of change of rate of change of a function. or we can say it is slope of slope. In simple word: Double differentiation is rate of change of slop. Geometrically: A second derivative is used to determine concavity. If you get a positive, it tells you that the graph ‘opens up’

When would the second derivative test be inconclusive?

0
If f′(c)=0 and f″(c)=0, or if f″(c) doesn’t exist, then the test is inconclusive.

What does double derivative represent?

The second derivative measures the instantaneous rate of change of the first derivative. The sign of the second derivative tells us whether the slope of the tangent line to f is increasing or decreasing.

What does it mean when the first derivative of a function is zero?

The first derivative being zero means that the function has a stationary value. The second derivative being zero means that you will have to work a little harder to decide whether it is a local maximum, minimum, or point of inflection.

When is the double derivative of a graph zero?

A double derivative is zero for a specific x value at inflection points (where the CHANGE of slope itself changes from positive to negative or negative to positive). At these points however, the slope (first derivative) should not be zero.

What is the difference between first and second derivative?

First derivative = 0. ==> function is optimum at the point. (a maximum or a minimum). Second derivative = < 0 ==> function has a maximum at the point. Second derivative = > 0 ==> function has a minimum at the point.

What is the second derivative test for critical points?

The point x may be a local maximum or a local minimum, and the function may also be increasing or decreasing at that point. The three cases above, when the second derivative is positive, negative, or zero, are collectively called the second derivative test for critical points.