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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.$$f(x)=6 x^{5}-5 x^{6}$$

Calculus 1 / AB

Chapter 3

Applications of the Derivative

Section 3

Concavity and the Second Derivative

Derivatives

Missouri State University

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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Find the $x$ - and $y$ -in…

questions. 60 would like you to use the 1st and 2nd derivative to grab F of X equals six x to the fifth minus five X to the sixth. Ah, and then you can use any intercepts to also help you graph it. If it's easy starting off with finding ah, the X axis intercepts, you can set this equal to zero. Factoring out X to the fifth euro equals X to the fifth times six minus five x So why is equal to zero at X zero and also at 6/5? Taking F prime of X. We can find critical points so 30 x to the fourth, minus 30 x to the fifth, setting that equal to zero. You have 30 x to the fourth times. One minus X is equal to zero, so X is equal to zero and one plugging that into F F of 00 and F one is one. Now, using F double prime of X, we can find common cavity. So after the prime of X is equal to 1 20 x cubed minus 1 50 x to the fourth F double prime of zero is zero, which means this is an inflection point. F double prime of one is negative. 30. So you have a concave down, which means it's a max. Um, and then also just testing some points around these numbers. So f double prime of negative 0.5 is equal to negative 5.6. So it's concave down after the crime of 0.5 is equal to 5.6. Therefore, we are concave up here. So now drawing this using all this information, we have 00 Yeah, And at 6/5, also a zero. Um, you also know that at 11 you have a point. So when X is less than zero, you're gonna be concave down between zero and one. Your concave up. This is your inflection point. Ah, after that, your concave down again. So this is what your graph looks like. And that's your answer to question 60

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