The number of solutions of the equation \(2\mathrm{x}+3\mathrm{tanx}=\pi ,\mathrm{x}\in [-2\pi ,2\pi ]-\left{\pm \frac{\…

Explanation

\(f(x)=2x+3\tan x-\pi\)

\(f'(x)=2+3\sec^2x\)

\(\sec^2x>0\),

\(f'(x)>0\)

for every point in the domain.

Therefore \(f(x)\) is strictly increasing on each interval

\(\left(-2\pi,-\frac{3\pi}{2}\right)\),

\(\left(-\frac{3\pi}{2},-\frac{\pi}{2}\right)\),

\(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\),

\(\left(\frac{\pi}{2},\frac{3\pi}{2}\right)\),

\(\left(\frac{3\pi}{2},2\pi\right)\).

Now check sign changes.

\(\left(-2\pi,-\frac{3\pi}{2}\right)\),

\(f(-2\pi)=-5\pi<0\),

\(f(x)\to+\infty\) as \(x\to\left(-\frac{3\pi}{2}\right)^-\).

Exactly one solution.

\(\left(-\frac{3\pi}{2},-\frac{\pi}{2}\right)\),

\(f(x)\to-\infty\) as \(x\to\left(-\frac{3\pi}{2}\right)^+\),

\(f(x)\to+\infty\) as \(x\to\left(-\frac{\pi}{2}\right)^-\).

Exactly one solution.

\(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\),

\(f(x)\to-\infty\) as \(x\to\left(-\frac{\pi}{2}\right)^+\),

\(f(x)\to+\infty\) as \(x\to\left(\frac{\pi}{2}\right)^-\).

Exactly one solution.

\(\left(\frac{\pi}{2},\frac{3\pi}{2}\right)\),

\(f(x)\to-\infty\) as \(x\to\left(\frac{\pi}{2}\right)^+\),

\(f(x)\to+\infty\) as \(x\to\left(\frac{3\pi}{2}\right)^-\).

Exactly one solution.

\(\left(\frac{3\pi}{2},2\pi\right)\),

\(f(x)\to-\infty\) as \(x\to\left(\frac{3\pi}{2}\right)^+\),

\(f(2\pi)=3\pi>0\).

Exactly one solution.

Total number of solutions

\(=1+1+1+1+1=5\).