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Use the second derivative test to classify the critical points.$$f(x)=3 x^{4}+16 x^{3}+6 x^{2}-72 x+20$$

$$\mathrm{m}(-3,101), \mathrm{M}(-2,108), \mathrm{m}(1,-27)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Harvey Mudd College

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.

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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Use the second derivative …

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question 42 would like you to use the second derivative test to classify critical points for F of X equals three x to the fourth plus 16 x cubed plus six X squared minus 72 X plus 20. 1st you need to find f prime of X to identify those critical points. So F prime of X is equal to 12 x cubed plus 48 x squared plus 12 ax minus 72. Uh, and then you need to factor this, uh, you can first test any number is divisible by negative 72. Um, I chose F prime of just one, and that did come out to zero, which tells you that X minus one can be factored out. From there. You can try and figure out what the coefficients of X squared X and your constant will be at the end here. You know that you're gonna be ending up with negative 72 and you're going to have 12 x cubed up here, which means that for X Times, X squared would need to be 12 x squared 12 X squared times. Negative one is negative 12 X squared and for you need your X squares to add to uh to 48 so this needs to be 60 x squared and which means that this would be just 60 x, the negative 60 x and your constant would need to multiply 2 72. Therefore just 72 which means 72 x, which comes out correct for your 12 x. Therefore, your factored F prime of X is X minus one times 12 x squared plus 60 X plus 72 Ah, then you can further factor this by taking out 12 so X minus one times 12 x squared plus five x plus six So then you're just left with X minus one 12 exposed to and expose three. Therefore, you're zeros here. Our X equals one negative two and negative three. Those are your critical points plugging them into f of X. You would have won negative 27 negative 208 and negative 31 Oh one. From there, we can now test con cavity of these points, So finding F double prime of X, that would just be 36 x squared plus 96 x plus 12 f double prime of one is equal to 1 44 which is concave up, and that means it's a minimum. So one negative 27 is a minimum and F double prime of negative to negative 36 concave down there's gonna be a max, and after the prime of negative three is 48. So you're concave up again, which means it's a minimum, and those are your answers for Question 42.

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