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For what values of $c$ does the polynomial $P(x)=x^{4}+c x^{3}+x^{2}$ have two inflection points? One inflection point? None? Illustrate by graphing $P$ for several values of $c .$ How does the graph change as $c$ decreases?

As the value of $c$ increases, one of the relative minimum points (the one onthe left $)$ dips lower and lower while the other (the one on the right) becomescloser and closer to the $x$ -axis.

Calculus 1 / AB

Chapter 4

APPLICATIONS OF DIFFERENTIATION

Section 3

Derivatives and the Shapes of Graphs

Derivatives

Differentiation

Applications of the Derivative

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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.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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for this program, we can take a second dirt your first. So we have 12 eggs square past six c x class to this is a quadratic function so far for and then notice that the leading term has, um, a positive coefficient 12. That means if we plot this quadratic function, Pete up of prime mix on the coordinates, he has three different types, so either has to a solution. And one solution that means this curve Just touch, touch this X axis at a single point like this, or we have no solution. Case that means this curve doesn't touch the X axis. Um, so for the first case, we noticed that there are two x intercept here in the footballs off them, they are inflection point. So they are inflection points for, um points four p x. Just because up on the second of narrative changes his it's sign from positive here is positive and the two negative here and the front negative took positive. So we have to inflection points. But for the forelock for next two types, there's no inflection points because on the second, figurative is ice positive. So in this case, we want to know one. Um, the second narrative has two x intercept. So by the quadratic formula, we have of Peace Square minus four a c, which is Ah, 36 C square minus 99. So if this is strictly positive, then there there are two solutions. So we can see that if sees greater than route off 8/3 or sees strictly lesson minus root of a over three, they are doing to our inflection points. Otherwise, no, you inflection points.

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