Question

F(x)=-4x^3+15x^2-8x-3
b.) Determine the possible number of turning points of f(x).
# of possible turning points:?

Answer 1

Answer:

Number of possible turning points = 2

Step-by-step explanation:

Given polynomial function:

$f(x)=-4x^3+15x^2-8x-3$

To determine the possible number of turning points of the given function, we need to examine the degree of the polynomial.

The degree of a polynomial is the highest power of the variable in the polynomial. In this case, the highest power of x is 3, so the degree of f(x) is 3.

The possible number of turning points for a polynomial of degree n is at most (n - 1). In this case, the possible number of turning points for f(x) is at most 3 - 1 = 2.

Therefore, the function f(x) can have at most 2 turning points.

$\hrulefill$

Additional Information

The number of turning points can be found by differentiating the function and setting the derivative equal to zero, which will then give the x coordinates of any turning points.

Differentiate the function:

$f'(x)=-12x^2+30x-8$

Set the derivative equal to zero:

$\begin{aligned}-12x^2+30x-8&=0\\-2(6x^2-15x+4)&=0\\6x^2-15x+4&=0\end{aligned}$

Solve for x using the quadratic formula:

$x=\dfrac{-(-15) \pm \sqrt{(-15)^2-4(6)(4)}}{2(6)}$

$x=\dfrac{15 \pm \sqrt{225-96}}{12}$

$x=\dfrac{15 \pm \sqrt{129}}{12}$

Therefore, the x-coordinates of the turning points of f(x) are:

$x=\dfrac{15 -\sqrt{129}}{12},\quad x=\dfrac{15 +\sqrt{129}}{12}$

This confirms that the function f(x) has two turning points.