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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.$$f(x)=x^{3}-6 x^{2}+8 x$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Campbell 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 49 would like you to use the 1st and 2nd derivatives to sketch F of X equals extra third minus six X squared plus eight. Um, and then also include any intercepts where they're easily determined. So starting off you can find some intercepts. So setting up of X equal to zero Ah, and factoring out acsi of x Times X squared minus six X plus eight and factoring that will be able to x times X minus four times X plus two Her ex I'm sorry X minus two. Therefore, uh, are zeros here? It would be zero for and two or when y is equal to zero. Uh, now finding any critical points, F Prime X is equal to three X squared minus 12 X plus eight. And to factor this, you can just use the quadratic formula so X is equal to 12 plus or minus square root of 1 44 minus four. Nice three times eight over six that can be simplified. Two. X equals to plus or minus two square root 3/3. So those are your critical points, finding their Y values F of two plus two screw three over three is negative 3.79 and F to minus two square three. Over three is 3.7 Not so positive from there. You can then find F double prime of X, which is six X minus 12. Uh, testing these two points these two critical points to find out if their maximum or minimum f of two or F double prime of two plus two squared 3/3 is 6.928 which means it's concave up. So it's a minimum and F double prime of two minus two square 3/3 is equal to negative. 6.928 concave down, which means it's a maximum now, taking those points Ah, and graphing them. Starting off, we can just put our zeros here. So 00 20 and 40 on the X axis and the Y axis here using our points two minus two squared three. Is that positive? Around? Positive three. And at around negative. Three. You've got two plus two Scroll three. Therefore, um, knowing that this is a maximum and this is a minimum, you can draw your graph like so connecting those points. And that's your answer to Question 49

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