Please explain graphically the above changes(I imply changes in roots, derivatives at some points, continuity and differentiability at some points etc

  • $\begingroup$ I cannot see a question here. $\endgroup$
    – Martin R
    Jan 7, 2018 at 18:52
  • $\begingroup$ I mean any function not very specific, I request you to answer my query using various functions as examples. $\endgroup$ Jan 7, 2018 at 18:54

2 Answers 2


Take the function $f(x) = 1$ if $x$ is rational and $-1$ if $x$ is irrational. Then $f$ is no where differentiable or continuous, and its graph has no tangent lines anywhere.

But $f^2(x) =1 $ for all real $x$, so it's differentiable and continuous everywhere and has a tangent at every point.

  • $\begingroup$ Can this be shown graphically? $\endgroup$ Jan 7, 2018 at 19:03
  • $\begingroup$ It's pretty easy to graph the function $y=1$ and observe its various properties. $\endgroup$
    – B. Goddard
    Jan 7, 2018 at 19:36

If y is f(x) then

$$(y^2)^{'} = 2 y y^{'}$$

The place where max/min occurs (x coordinate) is unaffected. But max / min of $y$ is itself changed as do others mentioned in title line.

  • $\begingroup$ The place where max/min occurs implies x coordinates, right? $\endgroup$ Jan 7, 2018 at 19:07
  • $\begingroup$ Yes, x coordinate. $\endgroup$
    – Narasimham
    Jan 7, 2018 at 19:10
  • $\begingroup$ Hey, impressed by your answer: math.stackexchange.com/a/1404664/510458 Please tell me which book: Higher algebra or elementary algebra.Tell the chapter too if you remember. $\endgroup$ Jan 7, 2018 at 19:15
  • $\begingroup$ Higher Algebra: A Sequel to Elementary Algebra for Schools Henry Sinclair Hall, Samuel Ratcliffe Knight; In India, standard algebra text in colleges for long time. $\endgroup$
    – Narasimham
    Jan 7, 2018 at 19:33
  • $\begingroup$ Got it.Thankyou. $\endgroup$ Jan 9, 2018 at 8:20

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