Question: Lesson: 6.4 The Fundamental Theorem of … Question 3 of 9, Step 4 of 4 5/15 Correct Consider the…

Lesson: 6.4 The Fundamental Theorem of … Question 3 of 9, Step 4 of 4 5/15 Correct Consider the following polynomial function. f(x) = x^4 + x^3 - 7x^2 - 5x + 10 Step 4 of 4: Find the zero(s) at which f” flattens out. Express the zero(s) as ordered pair(s). Select the number of zero(s) at which f” flattens out. Selecting an option will display any text boxes needed to complete your answer. none 1 2 3 4 Zero(s) at which f” flattens out:

Solution

To find the zero(s) where the function \( f(x) = x^4 + x^3 - 7x^2 - 5x + 10 \) “flattens out,” we need to determine the points where both the function and its derivative are equal to zero. Step 1 Find the first derivative of the function \( f(x) \). \[ f’(x) = 4x^3 + 3x^2 - 14x - 5 \] Step 2 Set both \( f(x) \) and \( f’(x) \) equal to zero to find the critical points. \[ \begin{cases} x^4 + x^3 - 7x^2 - 5x + 10 = 0 & \\ 4x^3 + 3x^2 - 14x - 5 = 0 & \end{cases} \] Step 3 Solve the system of equations to find the values of \( x \) that satisfy both equations. After solving the system, the solutions are: \[ (-2, 0) \\ (1, 0) \\ (2, 0) \] These ordered pairs represent the zero(s) where the function \( f(x) \) flattens out.