Question

How many relative extrema can the polynomial t(x)=3x2−5x+8x3
have?(1 point)
at most

Answer 1

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.