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Use the first and second derivatives to sketch the graph of the given equation. Also include the intercepts, whenever they are easily determined.$$f(x)=1 / 4 x^{4}-2 x^{2}$$

other zeros: $x=\pm 2 \sqrt{2}$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Oregon State University

University of Michigan - Ann Arbor

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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Use the first and second d…

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question 57 would like you to use the 1st and 2nd derivative to sketch the graph of f of X equals 1/4 X to the fourth minus two x squared, uh, and then include any intercepts if they're easy to find. So first finding the intercepts, um, on the X axis by setting this equation equal to zero, factoring out 1/4 x squared times X squared minus eight. Therefore, uh, you can say that X is equal to zero and plus or minus square root of eight square debate is about 2.8. So plus or minus 2.8 for when we graph taking f prime of X to find any critical points that be X to the third minus four x setting that equal to zero x times X squared minus four is equal to zero, therefore, X zero negative two and two. Now using F double prime of X, we can, uh, see if those are maxes or men's. That would be three X squared minus four. So F double prime of negative two is eight, which means concave up. It's a minimum F double prime of zero is negative four. So it's concave down a maximum n f double prime of two is eight. So as Khan gave up for a minimum to also find what these exact points are, we can take f of negative two, which is negative for F zero is zero and F of two is also negative. For so now plotting our points on our graph here going from negative 323 R X axis and going down to negative four on R Y axis. We know we have a point at 00 and at negative 24 we have our minimums into negative four, um, and then at 2.8 and negative, 2.8 and 2.80 we have our X axis intercepts. Therefore, knowing these two bottom points are a minimum, you can draw our function heading to our maximum back down to the minimum and back up. And that's your answer to question 57

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