Where does the second derivative equal zero?
Inflection points
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 if first and second derivative is 0?
When x is a critical point of f(x) and the second derivative of f(x) is zero, then we learn no new information about the point. The point x may be a local maximum or a local minimum, and the function may also be increasing or decreasing at that point.
What does it mean if derivative is zero?
Note: when the derivative curve is equal to zero, the original function must be at a critical point, that is, the curve is changing from increasing to decreasing or visa versa.
What does it mean when the second derivative does not exist?
If the limit (f'(x+h)-f'(x))/h as h->0 is undefined anywhere in the domain then f’ is not differentiable, and we could say f” does not exist (or at least that no function is the second derivative of f everywhere in its domain).
What happens if the first derivative is 0?
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.
Where is the derivative zero?
For what value(s) of x is the derivative zero? Answer: The sign of the derivative for the function is equal zero at the minimum of the function. The derivative is zero when x = 0.
Can an inflection point be zero?
An example of a stationary point of inflection is the point (0, 0) on the graph of y = x3. The tangent is the x-axis, which cuts the graph at this point. An example of a non-stationary point of inflection is the point (0, 0) on the graph of y = x3 + ax, for any nonzero a.
Can a point of inflection where the second derivative does not exist?
A point x=c is an inflection point if the function is continuous at that point and the concavity of the graph changes at that point. And a list of possible inflection points will be those points where the second derivative is zero or doesn’t exist.