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# Prove Taylor's Inequality for $n = 2,$ that is, prove that if $\mid f''' (x) \mid \le M \text { for }\mid x - a \mid \le d,$ then $\mid R_2 (x) \mid \le \frac {M}{6} \mid x - a \mid^3$ for $\mid x - a \mid \le d$

## $\left|R_{2}\left(x_{2}\right)\right| \leq \frac{1}{6} M|x-a|^{3}$

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##### Catherine R.

Missouri State University

##### Kristen K.

University of Michigan - Ann Arbor

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to prove the problem is proof Taylor's inequality for in musical tube. That is true that if there are derivative ABS, value is less than I am for absolutely. Of Oxman's A is less than the then at two x soiree of tracks is less than m over six times as value of Xmas eight to the power of three. Mm. As word of axeman is a is last D. So from here behalf ex prime from from X This is the last time, um, for X is greater than a and the last time they plus the than behalf integral from a two x What exit is spirit between A and a plastic us from from from two. He this last time integral from eight X Um, mhm. So on behalf years, this is of Hank's from from minus half A from prom is less than X months a and, um then we have integral of have x cramp from from h x r b u T t his last integral from a two x have from from okay class. Um, he minus a it's so we have at the tax prom minus have a problem is less than from? From a hams X minus. Mm. Class, Um, over two times X minus. Hey, square on behalf integral of I ftp from from eight x did He is less than 18 year old from a two ex prime a as I have from from eight times three months. Eight plus I'm over two times. He might stay square tiki than behalf have x minus f a. His life must, uh, have a prime times x minus a us f a from from 8/2 times Xmas a square us mm over six times X minus A to the power of three. Since our two X is equal to our fax minus f a minus, have a prom xmas a plus minus five apron, prime over two times X space square sofa. Half our two acts his last time I'm over six x minus A skill Uh huh. X is greater than a and less than plastic. A similar argument using us from from from is greater than negative arm. When X is between negative 50 plus eight and a we can prove r two X is greater than negative of three factorial times. Thanks. One stay. Q. So we have from these two results and the value of our two this last time. M over six times, absolutely of X minus Space cube.

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##### Catherine R.

Missouri State University

##### Kristen K.

University of Michigan - Ann Arbor

Lectures

Join Bootcamp