Quick Answer: What Happens When The First Derivative Is Undefined?

Can inflection points be undefined?

A point of inflection is a point on the graph at which the concavity of the graph changes.

If a function is undefined at some value of x , there can be no inflection point.

However, concavity can change as we pass, left to right across an x values for which the function is undefined..

Can a critical number be undefined?

A critical number for a function is any number in the function’s domain that causes the function’s first derivative to equal zero OR to be undefined. f`(x) is not defined for x = -2 or x = 2; however, -2 and 2 are not in the domain of function f.

Can derivatives be zero?

The derivative f'(x) is the rate of change of the value of function relative to the change of x. So f'(x0) = 0 means that function f(x) is almost constant around the value x0. … All these functions are almost constant around 0, which is the value where their derivatives are 0.

What happens when 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.

What happens when the first and second derivative is 0?

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 2nd derivative tell us?

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. … In other words, the second derivative tells us the rate of change of the rate of change of the original function.

How do you know if a derivative exists?

According to Definition 2.2. 1, the derivative f′(a) exists precisely when the limit limx→af(x)−f(a)x−a lim x → a f ( x ) − f ( a ) x − a exists. That limit is also the slope of the tangent line to the curve y=f(x) y = f ( x ) at x=a.

What is the first derivative rule?

If f′(x) changes from positive to negative at c, then f(c) is a local maximum. If f′(x) changes from negative to positive at c, then f(c) is a local minimum. If f′(x) does not change sign at c, then f(c) is neither a local maximum or minimum.

What happens when second derivative is undefined?

The second derivative is undefined at x=4, but this doesn’t negate the possibility of being concave down. The function is concave down if the derivative is decreasing. … Checking the second derivative is a test for concavity. If the second derivative does not exist, the test does not apply.

What if the derivative is undefined?

If there derivative can’t be found, or if it’s undefined, then the function isn’t differentiable there. So, for example, if the function has an infinitely steep slope at a particular point, and therefore a vertical tangent line there, then the derivative at that point is undefined.