The equation tan(x) = 3x^2 can be solved using numerical methods such as the Newton-Raphson method or the bisection method. However, it is not possible to find the exact solution of this equation using algebraic methods.
To determine the interval for which a root exists, you can use the intermediate value theorem.
First, observe that the left-hand side of the equation, tan(x), is undefined for x = (n + 1/2) π, where n is an integer. Thus, we can restrict our attention to the interval (-π/2, π/2) where the tangent function is continuous and strictly increasing.
Next, note that tan(0) = 0 and tan(π/6) = 1/√3 < 3/36 = 1/12. Also, as x approaches π/2 from the left, tan(x) approaches infinity, while 3x^2 approaches infinity faster. Therefore, there exists at least one root of the equation in the interval (0, π/6).
Similarly, tan(-π/6) = -1/√3 > -1/12, and as x approaches -π/2 from the right, tan(x) approaches negative infinity, while 3x^2 approaches infinity faster. Therefore, there exists at least one root of the equation in the interval (-π/6, 0).
Therefore, the equation tan(x) = 3x^2 has at least one root in the interval (-π/6, π/6).