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Determine where the function is concave upward and downward, and list all inflection points.$$f(x)=x \sqrt{3-x}$$

CD on $x<3$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Campbell University

Harvey Mudd College

Baylor University

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.

01:11

Determine where the functi…

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Determine the intervals on…

02:42

We want to identify any inflection points of the function F x equals X times square three minus x. Couldn't defy inflection points. We look for places where the second derivative F double prime changes sides from positive, negative or negative to positive. That means we have to find the second derivative. So F prime is three times x minus 2/3 minus x. F double prime is three times X minus 4/4 times three minutes. Actually three halfs to identify, whereas double prime can change sign. We have to look at this critical repetition points. Thus we look at the critical petition points as where the numerator denominator equal zero from the numerator of X equals four. For the denominator with X equals three. Those are critical repetition points are simply X equals three. The reason for this is the fact that from the function F X times route three minutes X. We have that X cannot be greater than three. This is because we can't take the spirit of a negative number. Thus we only have a critical point at three and we can want to evaluate the sign of F double prime. The left of X equals three. So let's practical three, such as X equals zero. We have that double prime is three times negative, four over four times three to the three halves. That is essential. Problem is negative. Thus we have that there's concave down From -1 of Eur three. Yeah, concave up nowhere and there are no inflection points because there's no point at which F double prime changes. Sign X ends at X equals three.

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