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Show that the second derivative test fails for: (a) $f(x)=(x-2)^{3}$ $$\text { (b) } f(x)=(x-2)^{4},\left(\text { c) } f(x)=-(x-2)^{4} ;(d)\right.$$ $f(x)=(x-1)^{3}(x+3)^{4}$ In each case, classify the critical point(s).

(a) neither(b) $m(2,0)$(c) $M(2,0)$(d) neither at $(1,0), M(-3,0), m(-5 / 7,-137.511)$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Missouri State University

Campbell University

Harvey Mudd College

University of Nottingham

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.

07:17

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01:07

For the final problem, we want to show that the second derivative test fails for fx Equalling X -2 Cubed. Uh So what this is gonna look like is we'll take the first derivative um and that's going to be three times x minus two squared and the second derivative is going to be um six Times X -2. So we see that the khan cavity test, the second derivative test is going to fail, because what we see is that this is going to be the inflection point is concave down this conclave up um and we see that it's reflected here. And the problem, if we look at the first derivative, though, we end up seeing that we end up getting this right here, so indicates that the graph is just increasing the whole time.

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