Question

How many relative extrema can the polynomial t(x)=3x^2−5x+8x^3 have?(1 point)

__at most

Answer 1

[AI-generated answer — Bot GPT 3.5]

At most, the polynomial t(x) can have 2 relative extrema.

Answer 2

[AI-generated answer — Step-by-Step Bot]

To find the relative extrema of a polynomial, we need to take the derivative of the polynomial and set it equal to zero. Then, we can solve for the values of x at which the derivative equals zero.

Given the polynomial t(x) = 3x^2 - 5x + 8x^3, let's find its derivative:

t'(x) = d/dx(3x^2 - 5x + 8x^3)
= 6x - 5 + 24x^2

To find the values of x where t'(x) = 0, we set t'(x) equal to zero:

6x - 5 + 24x^2 = 0

Now, we can solve this equation for x. However, this equation is a quadratic equation, so we might have up to two values for x where the derivative equals zero.

Using any suitable method of solving quadratic equations, we can find the solutions.

By factoring, the equation can be rewritten as:

(2x - 1)(12x + 5) = 0

Setting each factor equal to zero:

2x - 1 = 0 or 12x + 5 = 0

Solving each equation separately:

2x - 1 = 0
2x = 1
x = 1/2

12x + 5 = 0
12x = -5
x = -5/12

Therefore, the values of x where t'(x) = 0 are x = 1/2 and x = -5/12.

These values represent the potential relative extrema of the polynomial. The polynomial t(x) can have up to two relative extrema.