Real roots of exponential equation

[man111]

Number of roots of the equation [tex]4\cos(e^x) = 2^x+2^{-x},[/tex] Given [tex]\ln 2\pi<\log_{2}(2+\sqrt{3})[/tex]

[nevrodermit]

Let [tex]f(x)=4\cos{(e^x)}[/tex] and [tex]h(x)=2^x+2^{-x}[/tex]. Then

[tex]h'(x)=\ln{(2)}(2^x-2^{-x})[/tex]. It is zero when [tex]2^{2x}=0\Leftrightarrow x=0[/tex]. It is easy to check that [tex]h\downarrow x<0[/tex], [tex]h_{min}=h(0)=2[/tex], [tex]h \uparrow x>0[/tex]. Note that if [tex]f(x)=h(x)[/tex] has a solution than we need [tex]h(x)\le 4[/tex]. Setting [tex]y=2^x[/tex] we get [tex]y+y^{-1}\le 4\Rightarrow y\in [2-\sqrt{3}, 2+\sqrt{3}][/tex] and hence [tex]x\in [\log_{2}{(2-\sqrt{3})}, \log_{2}{(2+\sqrt{3})}][/tex]. Clearly, [tex]h(\log_{2}{(2\pm \sqrt{3}))}=\pm 4[/tex].

[tex]f'(x)=-4e^x\sin{(e^x)}[/tex]. It is zero when [tex]e^x=\pi k, k\in \mathbb{Z}[/tex]. Clearly, [tex]e^x \uparrow[/tex] and hence [tex]0.14957 < \pi k <6.68568[/tex] which means [tex]k\in \{1,2\}[/tex]. It easy to check that [tex]f\downarrow x<\ln{\pi}[/tex] and [tex]f_{min}=f(\ln(\pi))=-4[/tex] and that [tex]f\uparrow x\in [\ln(\pi), \ln(2\pi)][/tex] and that it increases for [tex]x>\ln(2\pi)[/tex] and also [tex]f_{max}=f(\ln(2\pi))=4[/tex].

Additionally, we need to calculate some values: [tex]f(\log_{2}{(2-\sqrt{3})})=3.95534[/tex], [tex]f(\log_{2}{(2+\sqrt{3})})=3.68034[/tex], [tex]h(\ln(\pi))=2.66332[/tex], [tex]h(\ln(2\pi))=3.85457[/tex], [tex]f(0)=4\cos(1)=3.99[/tex], [tex]h(0)=2[/tex].

Now, using the increase/decrease properties and the given intervals we conclude there is one root in each of the following intervals: [tex][\log_{2}{(2-\sqrt{3})},0][/tex], [tex][0,\ln(\pi)][/tex], [tex][\ln(\pi), \ln(2\pi))][/tex], [tex][\ln(2\pi)), \log_{2}{(2+\sqrt{3})}][/tex].

I will explain for the first root, for the other intervals the logic is analogous.
The interval is [tex][\log_{2}{(2-\sqrt{3})},0][/tex], both [tex]f,h[/tex] are decreasing in it and [tex]f(\log_{2}{(2-\sqrt{3})})<h(\log_{2}{(2-\sqrt{3})})[/tex] and [tex]f(0)>h(0)[/tex] so they must cross just one time.