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!

Implications of the tangent of a function being at a maximum

  1. Mar 3, 2014 #1
    Does the tangent of a function being at a maximum necessarily mean that the function itself is at a maximum?

    I am supposed to find whether del is at a maximum at w = (tansig*taneps)^(-1/2)

    del = arctan(w*(tsig-teps)/(1+(w^2*(tsig*teps))))

    tansig and taneps are constants and w is the independent variable

    Using MATLAB, I've found that the derivative of tan(del) at the given w is in fact 0, and using a given graph of tan(del), I can see that the only point where the slope of the tangent is 0, is at a maximum peak. And so, I know that tan(del) at the given w is maximum.

    Knowing this, can I claim that del itself is maximum at the specified w?

    Thanks you all!
     
  2. jcsd
  3. Mar 3, 2014 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    Not clear what you mean by that - the maximum possible slope for a tangent is infinity (a vertical line).

    If f(x) has a maximum at x=a then the slope of the tangent at a is f'(a)=0
    But that is also true if f(x) has a minimum or a point of inflexion at x=a.

    Guessing that you are asking if the maxima of f(x) is also a maxima of f'(x)....
    consider:

    f(x)=4-x^2 has a maximum at x=0

    the slope of the tangent of f at x is: f'(x) = -2x

    f'(x) has a (possible) maxima when f''(x)=0 but f''(x)=-2 - so it cannot be zero!

    So you see that the location of the maxima of f(x) cannot be a maxima of f'(x).

    The second derivative does give you a clue though.
     
  4. Mar 3, 2014 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    I'll use the symbols ##b## and ##a## instead of tsig and teps, resp., so your function is
    [tex]f(w) = \arctan\left(\frac{(b-a)w}{1+ab\,w^2}\right) [/tex]

    If ##a## and ##b## are both > 0 then the point ##w_0 = 1/\sqrt{ab}## is, indeed, a stationary point. Whether is is a maximum or a minimum must be checked by a second-order test: ##w_0## is a maximum if ##f''(w_0) < 0## and is a minimum if ##f''(w_0) > 0##, where ##f''(w)## is the second derivative of ##f(w)##.

    If ##a## and ##b## have opposite signs the function ##f(w)## has no stationary points, so has no points where the tangent line is horizontal. Nevertheless, it has finite maxima and minima!
     
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: Implications of the tangent of a function being at a maximum
  1. Maximum of a function (Replies: 5)

  2. Maximum of a Function (Replies: 2)

Loading...