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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.$$f(x)=x^{4}-24 x^{2}$$

$$m \text { at }(\pm 2 \sqrt{3},-144)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Baylor University

University of Nottingham

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 first and second d…

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question 51 would like you to use the 1st and 2nd derivatives to draw a graph of f of X equals X to the fourth minus 24 x squared, Uh, then including the intercepts If it's easy to find, uh, starting off, we can just find intercept. So, uh, factoring out, I squared, you get X squared minus 24 which means, uh, why is equal to zero at X equals zero and plus or minus square root of 24. From there, we can find critical points by taking F prime of X is equal to four x cubed minus 48 X critical points would be when this is equal to zero. So four X times X squared minus 12 is equal to zero at X equals zero and plus or minus square root of 12. From there, we can find what those values are by plugging them into F. So f zero zero, a negative square root of 12. His negative 1 44 and F of square root 12 is equal to thank positive. Oh, also negative. When I'm 34 from there, then we can get F double private X to determine if these are max or minimums. So after the prime of X is equal to 12 x squared minus 48 therefore F double prime of zero is negative 48 which means it is concave down. So a max F double prime of negative screw 12 is equal to 96. So concave up minimum and after a prime of screw 12 is also 96 which means it's also concave up, and it's also a minimum. So now, using all the information we can draw a graph which goes screw 12 is roughly 3.5 and scored 24 is roughly 4.9. So using that to draw our access here five two negative five on the X axis and down to negative 1 44 on R Y axis, we can then plot some of our 0.0 Ah, negative skirt 12 and negative 1 44. That is roughly here and again on the other side. Native 1 44 at Positive Square Root 12. Um, and you also have an intercept at about 4.9 and negative 4.9. Therefore, uh, knowing that these two are minimum. And at 00 is your maximum should look something like this as you connect your points together and that is your answer to question 51.

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