A ques on trig max min values

[maths_hero]

I've been asked to find the max and min values of the following expression:


I know how to find the max min values of the expressions of the following type:
Acosx+ Bsinx, extreme values:+-√ (A²+B²)

How to express the the value in the question to some form of Acosx+ Bsinx or what should be my approach?

 

[Dr.Peterson]

I assume this isn't a calculus question? That's how I'd normally want to approach it.

Because you have trig functions of both x and 2x, they will not combine (unless something special happens) to make a single trig function of x, or of 2x. This is like combining two harmonics of the same note, which produces a more complicated waveform.

You can write it in the form [MATH]a\sin(x + b) + \sin(2x)[/MATH], as you are suggesting, but I don't think that helps algebraically. At that point, I'd perhaps just sketch it and hope things align such that you can tell where the maximum will be.

In fact, when I try it, that is exactly what happens. You will be able to use symmetry to convince yourself that what looks like the max and min are in fact the max and min (though the min is a little less convincing to my mind -- and it turns out that a slightly smaller multiplier on the first term would change things).

The important thing is that this is a special case; in general, a question like this would require calculus or technology.

 

[maths_hero]

At that point, I'd perhaps just sketch it and hope things align such that you can tell where the maximum will be.

In fact, when I try it, that is exactly what happens. You will be able to use symmetry to convince yourself that what looks like the max and min are in fact the max and min (though the min is a little less convincing to my mind -- and it turns out that a slightly smaller multiplier on the first term would change things).

Well it worked out, I factorized the expression into 4sin(x+π/4) + sin2x and graphed it and got the max and min values at x=π/4 and x=-3π/4, never thought of this approach before. Thanks

 

[Dr.Peterson]

Yes, here is my graph:



And if we change the [MATH]2\sqrt{2}[/MATH] in the problem to 2, here is what happens, illustrating the specialness of the numbers:



Suddenly, the minimum isn't where you would have thought!

 

[maths_hero]

Yes, you're right this question was specifically designed for our standard, otherwise it will be hard to determine which is evident from the graphs you drawn.