How do you find the maximum of #f(x) = 2sin(x^2)#?

↳Redirected from "Suppose that I don't have a formula for #g(x)# but I know that #g(1) = 3# and #g'(x) = sqrt(x^2+15)# for all x. How do I use a linear approximation to estimate #g(0.9)# and #g(1.1)#?"
1 Answer
Jim H
Mar 6, 2015

The maximum value of #f(x)=2sin(x^2)# is #2#.

This is an example of a kind of problem sometimes asked in a calculus class. It is a kind of a trap question. You do not need calculus to answer this question.

The maximum value of the sine function is #1#. This happens when #x^2=pi/2+2 pi k# for any integer #k#.

I also know that #x^2= pi/2# when #x=sqrt(pi/2)#.

So the maximum value of this function is #2#.

If you need to know where else the maximum occurs, then you need to solve: #x^2=pi/2+2 pi k=pi/2 (1+4k)# for any integer #k#.

The solutions are #x=+-sqrt(pi/2)sqrt (1+4k)# which might be easier to read written: #x=+-sqrt (1+4k)sqrt(pi/2)# for any integer k.

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