# Second Derivatives and Beyond

# Second Derivatives and Beyond: Thinking Critical Quiz

Think you’ve got your head wrapped around

*? Put your knowledge to the test. Good luck — the Stickman is counting on you!***Second Derivatives and Beyond**Q. A critical point of a function

*f*is a point at which*f*' is zero

*f*' is zero or undefined

*f*" is zero or undefined

*f*" changes sign

Q. Which of the following statements about the function
is true?

The function

*f*has no critical points.*x*= 0 is the only critical point of

*f*.

*x*= 2 is the only critical point of

*f*.

*x*= 0 and

*x*= 2 are both critical points of

*f*.

Q. An inflection point of a function

*f*is a point at which*f*' is zero

*f*' is zero or undefined

*f*" is zero or undefined

*f*" changes sign

Q. Which of the following statements about the function

*f*(*x*) = 4(*x*- 1 ) - 8 is true?The function

*f*has no inflection points.*x*= 1 is the only inflection point of

*f*.

*x*= 3 is the only inflection point of

*f*.

*x*= 1 and

*x*= 3 are both inflection points of

*f*.

Q. A maximum value of a function

*f*isa

*y*-value that is bigger than every other value of*f*.a

*y*-value that is bigger than every nearby value of*f*.an

*x*-value at which*f*is bigger than anywhere else.an

*x*-value at which*f*is bigger than anywhere nearby.Q. How many minima does the function

*f*(*x*) = -5cos x have?none

one

finitely many, but more than one

infinitely many

Q.

*x*=*c*is a critical point of the function*f*. The sign of*f*' is negative to the left of*c*and positive to the right of*c*. Which of the following statements is true?*x*=

*c*is a maximum value of

*f*.

A maximum value of

*f*occurs at*x*=*c*.*x*=

*c*is a minimum value of

*f*.

A minimum value of

*f*occurs at*x*=*c*.Q.

*x*=*c*is a critical point of the function*f*. If*f*"(*c*) = 0, which of the following statements is true?The second derivative test says that a minimum value of

*f*occurs at*x*=*c*.The second derivative test says that a maximum value of

*f*occurs at*x*=*c*.The second derivative test says that neither a minimum nor a maximum value of

*f*occurs at*x*=*c*.The second derivative test says nothing at all about what occurs at

*x*=*c*.Q. If the derivative of a function

*f*is*f*'(*x*) = (*x*- 2 )(*x*+ 5), which of the following statements is true?*f*has a minimum at

*x*= 2 and a maximum at

*x*= -5.

*f*has a maximum at

*x*= 2 and a minimum at

*x*= -5.

*f*has minima at both

*x*= 2 and

*x*= 5

*f*has maxima at both

*x*= 2 and

*x*= 5

Q. The derivative of the function

*f*is*f*'(*x*) = 4*x* + 8

Which of the following statements is true?

*f*has a minimum at

*x*= 2.

*f*has a maximum at

*x*= 2.

*f*has a minimum at

*x*= -2.

*f*has a maximum at

*x*= -2.