91Ó°ÊÓ

First, we will compute the derivative of the given function, $$g(t)=\ln \left(3 t^{2}+1\right)$$ with respect to t. Using the chain rule, we get: $$g'(t)=\frac{d}{dt}\ln \left(3 t^{2}+1\right) = \frac{1}{3t^{2}+1}\cdot (6t) = \frac{6t}{3t^{2}+1}$$

Now, we will compute the second derivative \(g''(t)\). Apply the quotient rule, we get: $$g''(t)=\frac{d}{dt}\left(\frac{6t}{3t^{2}+1}\right) = \frac{(3t^{2}+1)(6)-(6t)(6t)}{(3t^{2}+1)^{2}}$$ Simplify the expression: $$g''(t)=\frac{18t^2+6-36t^2}{(3t^{2}+1)^{2}} = \frac{-18t^2+6}{(3t^{2}+1)^{2}}$$

Analyze the second derivative \(g''(t)\) to find the intervals where it is positive or negative. The denominator of \(g''(t)\) is always positive because of the square function. So, the sign of \(g''(t)\) depends on its numerator. The numerator is a quadratic function that opens downwards, so it is positive when \(-18t^2+6>0\), which is equivalent to: $$t^2<\frac{1}{3}$$ Thus, the function is concave up on the interval \((-\frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})\). On the other hand, \(g''(t)<0\) when \(t^2>\frac{1}{3}\). Hence, the function is concave down on the intervals \((-\infty, -\frac{\sqrt{3}}{3})\) and \((\frac{\sqrt{3}}{3}, \infty)\).

Inflection points occur when the second derivative changes its sign. In this case, there are two inflection points at \(t=-\frac{\sqrt{3}}{3}\) and \(t=\frac{\sqrt{3}}{3}\). To find the corresponding function values, substitute these points into the original function: $$g\left(-\frac{\sqrt{3}}{3}\right)=\ln\left(3\left(-\frac{\sqrt{3}}{3}\right)^2+1\right)=\ln(2)$$ $$g\left(\frac{\sqrt{3}}{3}\right)=\ln\left(3\left(\frac{\sqrt{3}}{3}\right)^2+1\right)=\ln(2)$$ So, the inflection points are \(\left(-\frac{\sqrt{3}}{3},\ln(2)\right)\) and \(\left(\frac{\sqrt{3}}{3},\ln(2)\right)\). In summary, the function is concave up on the interval \((-\frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})\), concave down on the intervals \((-\infty, -\frac{\sqrt{3}}{3})\) and \((\frac{\sqrt{3}}{3}, \infty)\), with inflection points at \(\left(-\frac{\sqrt{3}}{3},\ln(2)\right)\) and \(\left(\frac{\sqrt{3}}{3},\ln(2)\right)\).