1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Find a for which function f has no critical number

  1. Mar 25, 2017 #1
    1. The problem statement, all variables and given/known data
    Determine the values of the number ##a## for which the function ##f## has no critical number. $$f(x) = (a^2+a-6)\cos{2x} + (a-2)x + \cos{1}$$

    2. Relevant equations
    Concept of critical point

    3. The attempt at a solution
    Now the derivative of this function will be ##f'(x) = -2(a^2+a-6)\sin{2x} + (a-2)##. Now the amplitude of the sine function is ##2 |a^2+a-6|##. The maximum and the minimum values of the first term are ##2 |a^2+a-6|## and ##-2 |a^2+a-6|##. So ##f'(x)## will never be zero if ##|a-2| > 2 |a^2+a-6|##. And if ##f'(x)## is never zero, then the function ##f## will have no critical number. ##f'(x)## exists for all real numbers, so we don't have to worry about the points at which ##f'(x)## does not exist. So function ##f## will have no critical number if ##|a-2| > 2 |a^2+a-6|##. Solving this inequality, we get ##-\frac{7}{2} < a < -\frac{5}{2} ##. Is this correct ?
     
  2. jcsd
  3. Mar 25, 2017 #2

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Looks right. Not sure how you did the last step, but substituting a=c+2 makes it easy.
     
  4. Mar 26, 2017 #3
    haruspex, I considered cases here. Since ##(a^2+a-6) = (a-2)(a+3)##, I considered cases where ##a < -3##, and ##-3 < a < 2## and ##a>2##. Whenever there are abslute values, taking different cases help. But yes, with your substitution, things become little easier.
     
  5. Mar 26, 2017 #4

    haruspex

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Yes, it avoids the need to break into cases. |c|>2|c2+5c|, so 1>2|c+5|.
     
  6. Mar 26, 2017 #5
    Makes sense, haruspex
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Find a for which function f has no critical number
Loading...