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Worksheet: Rolle’s Theorem

Q1:

Consider the function 𝑓 ( π‘₯ ) = π‘₯ 𝑒 2 π‘₯ .

Determine when 𝑓 β€² ( π‘₯ ) = 0 .

  • A π‘₯ = 0 , π‘₯ = βˆ’ √ 2
  • B π‘₯ = 0 , π‘₯ = √ 2
  • C π‘₯ = 0 , π‘₯ = βˆ’ 2
  • D π‘₯ = 0 , π‘₯ = 2
  • E π‘₯ = 2 , π‘₯ = βˆ’ 2

Where on the number line is 𝑓 β€² ( π‘₯ ) < 0 ?

  • A ( βˆ’ ∞ , 0 ) βˆͺ ( 2 , ∞ )
  • B ( βˆ’ ∞ , ∞ ) βˆ’ ( 0 , 2 )
  • C ( βˆ’ ∞ , ∞ )
  • D ( 0 , 2 )
  • E [ 0 , 2 ]

What is l i m π‘₯ β†’ ∞ 𝑓 β€² ( π‘₯ ) ?

A sketch of 𝑦 = 𝑓 β€² ( π‘₯ ) on the interval [ 2 , ∞ ) is a curve below the π‘₯ -axis that is zero at π‘₯ = 2 and tends to zero as π‘₯ β†’ ∞ . Knowing that 𝑓 β€² is differentiable, what would an extended Rolle’s theorem tell us about 𝑓 on the interval ( 2 , ∞ ) ?

  • AThe function has an inflection point at some point π‘Ž ∈ ( 2 , ∞ ) , where 𝑓 β€² β€² ( π‘Ž ) = 0 .
  • BThe function is decreasing before some point π‘Ž ∈ ( 2 , ∞ ) then starts increasing after that point.
  • CThe function has a sharp corner at some point π‘Ž ∈ ( 2 , ∞ ) .
  • DWe cannot get any information about 𝑓 .
  • EThe function has a local minimum at some point π‘Ž ∈ ( 2 , ∞ ) .