Intermediate value theorem question

Question

Hi,

I need to show that there at least two values of x in the interal [0, pi] for which 3 sin x - 2 cos x = x

I think i need to use the Intermediate value theorem, but not sure how i would do this.

Any help would be much appreciated.

Thank You

Kara x

Answer 1

If you evaluate the function at x=0 you get y=3(0)-2(1)=-2 which is <x as x=-2

If you evaluate at x=pi/2you get 3(1)-2(0)=3 which is >pi/2= 1.57

since the function is continuous and it has gone from a value less than x to a value greater than x then it would have to pass through a point where the 2 are equal

If you do the same calculation at x=pi (I will leave this to you) you will see that y=3sinx -2cosx is <pi(3.14). So udon got the same logic it would again have to pass though a second point where they are equal.

So you have proven that there are at least 2 points where the two functions are equal.

You can see this graphically on Desmos

‘graph y= 3sinx-2cosx and y=x and you will see they cross twice in the interval (0 - pi)