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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.$$f(x)=1 / 4\left(x^{4}-4 x^{3}+12 x^{2}-8 x+11\right)$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Campbell University

Oregon State University

Baylor University

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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we went to graph alphabet X equals 1/4 times X. To the fourth minus four X. Cubed plus 12 X squared minus eight X plus 11. To do so we're gonna deploy calculus curve sketching techniques. So we're gonna analyze F. Prime and ethical crime successively before grabbing first for F. We get basic info on the graph. The domain is all X. There's no Y intercept or a sentence rex. We analyze F. Prime to see where the function is decreasing and increasing. F. Prime is execute minus three X squared plus six X minus two. There's a practitioner critical point X is approximately 20.0.4. So we have to evaluate prime. The left and right of 0.4 for the left, F. Prime is negative. So it is decreasing to the right of positive. So F. Is increasing. Finally we analyze ethical crimes working cavity purposes. After the problem is three times experiments to expose to. There's no protection of critical points. So we evaluate every time for all X. It's positive. so if it's concrete everywhere, thus we can graph F. We have our con cavity as an ethnic conflict everywhere, we have our minimum at .4, where we change from decreasing increasing.

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