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

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Baylor University

University of Michigan - Ann Arbor

University of Nottingham

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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question 46 would like you to use the 1st and 2nd directive to sketch the graph of F of X equals three X squared minus six X plus six. Ah, and then, if you can find any intercepts, include those. So to do so, F prime of X is equal to six x minus six. Any critical points would be when this is equal to zero. So when X is equal to one F double prime of X, then is equal to six. Which means that are critical. Point here is going to be concave up, so it's a minimum. Now plugging X equals one into f of X, half of one is three. So the 0.13 ah, then just testing for intercepts f of zero is equal to 606 and I'm just looking at F two is equal to six. So 26 therefore a drawing these points on our graph. You have 13 06 and to six, and we know this is a minimum. So our graph should look something like this. And that's your answer to question 46

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