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Prove that if $(c, f(c))$ is a point of inflection of the graph of $f$ and $f^{\prime \prime}$ exists in an open interval that contains $c$ , then $f^{\prime \prime}(c)=0 .$ [Hint. Apply the First Derivative Test and Fermat's Theorem to the function $g=f^{\prime} . ]$

$g=f^{\prime}(x)$ is differentiable on open interval containing $c .$ since $(c, f(c))$ is apoint of inflection, the concavity changes at $x=c .$Therefore $f^{\prime \prime}(x)$ changes sign at $x=c$ .Thus, by the First Derivative test, $f^{\prime}(x)$ has a local extremum at $x=c$Hence, by Fermat's theorem, $f^{\prime \prime}(c)=0$ q.e.d.

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

Campbell University

University of Michigan - Ann Arbor

Idaho State University

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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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Okay, so we couldn't do this. So the geometrically inflection point means that that the concave it is off their curve. It's changing. So supposedly Khalfan you fit. So for this example, we have an inflection point here because on its left, the functions can gave up. So the conflict, the fashions concave, come. And the only writes that functions can cave up. Um, so we can see there a small neighborhood, at least inflection points. We will see that, um, they exist to a point. For example, A and B at a point A and the point B, they have the same slope off that, uh, since low public attention line on them is So this is our observation. That means if we say for this, funky is a So if prime a graze on a and be so if we said you 55 he eggs, she would be f prime. So we have G A. Because the GP and by the um for most serum G prime has to be zero at some point, uh, between okay, and be so g Prime minister say conductivity for five. Um, so big. Basically, um, if so, by this argument, if g primary causes, eerie has to be the inflection points because we can choose a be arbitrarily

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