Given this equation $a\cos{x}+b=x$ with $a,b>0$ how to prove that there is at least one root between $(0,a+b]$ ?
For $x=0$ its $a+b$ which is >0
For $x=a+b$ its $a\cos(a+b) -a=a(\cos(a+b)-1)\leq0$
The problem here is that it is less than equal not less equal. Any idea what should I do ?
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GorillaApe
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2Why is that a problem? If it's equal to $0$, then there's your root. – Greg Martin Jan 27 '15 at 23:27
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cause theory has < . it makes sense but formally not – GorillaApe Jan 27 '15 at 23:40
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So split into cases. If it equals $0$, solve the problem by hand; if it doesn't equal $0$, use the theory. – Greg Martin Jan 28 '15 at 00:42