To find the maximum number of relative extrema that a polynomial can have, we can use the fact that relative extrema occur where the first derivative of the polynomial equals zero.
Let's start by finding the polynomial \( t(x) = 3x^2 - 5x + 8x^3 \). Combining the terms, we rewrite it as:
\[ t(x) = 8x^3 + 3x^2 - 5x \]
Next, we find the first derivative \( t'(x) \):
\[ t'(x) = \frac{d}{dx}(8x^3 + 3x^2 - 5x) = 24x^2 + 6x - 5 \]
This is a quadratic polynomial. A quadratic polynomial can have at most two real roots, which correspond to the critical points where the first derivative is zero. Each of these critical points can potentially be a relative maximum, minimum, or neither.
Therefore, the polynomial \( t(x) = 3x^2 - 5x + 8x^3 \) can have at most 2 relative extrema.