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Determine where the function is concave upward and downward, and list all inflection points.$$f(x)=x^{4 / 5}-2 x^{3 / 5}$$

CD if $x<0$ or $x>243$ CU if $0<x<243$ I (0,0) and (243,27)

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

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University of Michigan - Ann Arbor

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Boston College

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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Determine where the functi…

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Determine the intervals on…

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question 37 would like you to determine where f of X equals X to the fourth over. Five minus two X to the 3/5 is concave up or down and list any inflection points. So you need to find the second derivative. So starting off F prime of X is equal to for over five x to the negative. 1/5 minus two times 3/5 x to the negative 2/5 which is equal to 4/5. Excellent negative 1/5 minus 6/5 x to the negative 2/5. Um, now taking the driver again. So f double prime of X is equal two negative four over 25 x to the sixth negative 6/5 um, plus 12 over 25 x to the negative 7/5. From there, we can find inflection point. So when F double prime of X is equal to zero, um, to do so, set this equal to zero. So zero equals negative. 4/25 x to the 6/5 plus 12/25 next to the 7/5 for over 25 x to the 6/5 is equal to 12/25 x to the 7/5. From there, 12 or four is equal to those 25. Cancel out so X to the 7/5 over X to the 6th 5th. Therefore, three is equal to X to the 1/5 if you take everything to the 52. 5. expanded to five. You just get X is equal to 243 so that is one of your inflection points. Plugging that into just regular F of X half of 2 43 is equal to 27 so one of your inflection points is 243 and 27. Another place where khan cavity will change is when this is undefined. So that's undefined at X equals zero, because you have zero on the denominator. Technically, it's not an inflection point. It's innocent. Oat so at X equals zero. There's an assassin tote, but it's technically a place of inflection. So now testing con cavity when X is less than zero testing after well, prime. So after the prime negative one is negative. 10.64 so you're concave down when X is less than zero for in between zero and 243 f double prime of 50. It is 5.44 times 10 to the negative fourth, which is positive. So you're concave up between zero and to 43. And lastly, when X is greater than 243 f double prime of 2. 50 is negative 1.201 times 10 to the negative. Six. Therefore, your con cave down when X is greater than 243 and those are your answers to question 37.

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