121616949-math.115

# 121616949-math.115 - f x = x 3-x f ± x = 3 x 2-1 f ±± x...

This preview shows page 1. Sign up to view the full content.

5.4 Concavity and inflection points 101 a . a . Figure 5.6 f ( a ) < 0: f ( a ) positive and decreasing, f ( a ) negative and decreasing. down and is getting steeper. The two situations are shown in figure 5.6 . A curve that is shaped like this is called concave down. If we are trying to understand the shape of the graph of a function, knowing where it is concave up and concave down helps us to get a more accurate picture. Of particular interest are points at which the concavity changes from up to down or down to up; such points are called inflection points. If the concavity changes from up to down at x = a , f changes from positive to the left of a to negative to the right of a , and usually f ( a ) = 0. We can identify such points by first finding where f ( x ) is zero and then checking to see whether f ( x ) does in fact go from positive to negative or negative to positive at these points. Note that it is possible that f ( a ) = 0 but the concavity is the same on both sides; f ( x ) = x 4 is an example. EXAMPLE 5.7 Describe the concavity of f ( x ) = x 3 - x .
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: f ( x ) = x 3-x . f ± ( x ) = 3 x 2-1, f ±± ( x ) = 6 x . Since f ±± (0) = 0, there is potentially an inﬂection point at zero. Since f ±± ( x ) > 0 when x > 0 and f ±± ( x ) < 0 when x < 0 the concavity does change from down to up at zero, and the curve is concave down for all x < 0 and concave up for all x > 0. Note that we need to compute and analyze the second derivative to understand con-cavity, so we may as well try to use the second derivative test for maxima and minima. If for some reason this fails we can then try one of the other tests. Exercises Describe the concavity of the functions in 1–18. 1. y = x 2-x ⇒ 2. y = 2 + 3 x-x 3 ⇒ 3. y = x 3-9 x 2 + 24 x ⇒ 4. y = x 4-2 x 2 + 3 ⇒ 5. y = 3 x 4-4 x 3 ⇒ 6. y = ( x 2-1) /x ⇒ 7. y = 3 x 2-(1 /x 2 ) ⇒ 8. y = sin x + cos x ⇒ 9. y = 4 x + √ 1-x ⇒ 10. y = ( x + 1) / p 5 x 2 + 35 ⇒ 11. y = x 5-x ⇒ 12. y = 6 x + sin 3 x ⇒ 13. y = x + 1 /x ⇒ 14. y = x 2 + 1 /x ⇒...
View Full Document

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern