What\:is\:the\:smallest\:value\:of\:a\:such\:that\:the\:function\:\:\left(a,\:\infty \:\right)\Rightarrow \:R,\:g\left(t\right)=\frac{3t}{5+t^2}\:\:,\:has:an\:inverse\:function?

To determine the smallest value of \( a \) such that the function \( g(t) = \frac{3t}{5+t^2} \), defined on the interval \( (a, \infty) \), has an inverse function, we need to ensure that \( g(t) \) is one-to-one on this interval. This can be achieved by ensuring that \( g(t) \) is either strictly increasing or strictly decreasing on \( (a, \infty) \).

First, we compute the derivative of \( g(t) \):

\[
g'(t) = \frac{d}{dt} \left( \frac{3t}{5 + t^2} \right)
\]

Using the quotient rule:

\[
g'(t) = \frac{(5 + t^2) \cdot 3 - 3t \cdot 2t}{(5 + t^2)^2} = \frac{15 + 3t^2 - 6t^2}{(5 + t^2)^2} = \frac{15 - 3t^2}{(5 + t^2)^2}
\]

For \( g(t) \) to be strictly increasing or strictly decreasing, \( g'(t) \) should not change sign. This means \( \frac{15 - 3t^2}{(5 + t^2)^2} \) must be either always positive or always negative.

Set \( g'(t) = 0 \) to find the critical points:

\[
15 - 3t^2 = 0 \implies t^2 = 5 \implies t = \pm \sqrt{5}
\]

Thus, the function changes concavity at \( t = \pm \sqrt{5} \).

To ensure \( g(t) \) is one-to-one, we need \( g'(t) \) to be either positive or negative on the interval \( (a, \infty) \). We can choose the interval such that it does not include points where the derivative changes sign, such as \( t = \sqrt{5} \).

Consider the interval \( (\sqrt{5}, \infty) \):

- For \( t > \sqrt{5} \), \( t^2 > 5 \), and therefore \( 15 - 3t^2 < 0 \). So \( g'(t) < 0 \) in \( (\sqrt{5}, \infty) \).

Since \( g'(t) \) is negative in this interval, \( g(t) \) is strictly decreasing on \( (\sqrt{5}, \infty) \).

Thus, the smallest value of \( a \) such that \( g(t) \) has an inverse function is \( \sqrt{5} \).

**Answer:** The smallest value of \( a \) is \( \sqrt{5} \).

Would you like more details or have any questions?

Here are 5 questions you might want to ask:
1. How do you find the derivative of a function using the quotient rule?
2. What is the importance of a function being one-to-one in having an inverse?
3. How do critical points help determine intervals for monotonicity?
4. Can you provide an example of finding the inverse of a simple function?
5. How do we test if a function is strictly increasing or decreasing?

**Tip:** When determining if a function has an inverse, always check if it is one-to-one by analyzing its derivative.

Discover how to find the smallest value of 'a' for which the function g(t) = 3t / (5 + t^2) has an inverse function using calculus concepts. Learn about derivatives, monotonicity, and the importance of one-to-one functions in calculus.

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