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Suppose that $f^{\prime}(c)=f^{\prime \prime}(c)=0,$ but $f^{\prime \prime}(x)$ is negative (positive) just to the left and right of $x=c .$ Show that the function has a relative maximum (minimum) at $x=c .$ Use this result to classify the critical points in Exercise 71

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

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Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

30:01

In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (the rate of change of the value of the function). If the derivative of a function at a chosen input value equals a constant value, the function is said to be a constant function. In this case the derivative itself is the constant of the function, and is called the constant of integration.

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For the following problem, we want to show that f prime fc equals at double prime Fc, which equals zero. Um But F double prime of X is positive, whereas negative just to the left and right of X equal seat. So I want to show that the function has a relative minimum at x equal. See So since we know that if privacy is zero, we know that that means it has to be either a minimum or a maximum. Um And we know that and we have something like this, for example, um X crib but this is a local minimum and we see that the derivative, if this was F affects the derivative and then the second derivative we would have has a constant value. But we see that the slope is going to be zero. So based on this, we end up finding that um it's going to be a local minimum.

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