Answer :
To solve this problem, we need to find the zero(s) of the polynomial function [tex]\( f(x) = x^4 - 3x^3 - 17x^2 + 21x + 70 \)[/tex] where the function "flattens out." This means we're looking for points where both the first and second derivatives are zero.
Here's a step-by-step guide to find these points:
1. Find the First Derivative:
The first derivative of the function [tex]\( f(x) \)[/tex] will help us find the critical points. Let's calculate it:
[tex]\[
f'(x) = \frac{d}{dx}(x^4 - 3x^3 - 17x^2 + 21x + 70) = 4x^3 - 9x^2 - 34x + 21
\][/tex]
2. Solve for Critical Points:
Critical points occur where the first derivative is zero, [tex]\( f'(x) = 0 \)[/tex]. Solve the equation:
[tex]\[
4x^3 - 9x^2 - 34x + 21 = 0
\][/tex]
By solving this cubic equation, we can find the critical points.
3. Find the Second Derivative:
The second derivative will help identify where the function "flattens out." Calculate the second derivative:
[tex]\[
f''(x) = \frac{d}{dx}(4x^3 - 9x^2 - 34x + 21) = 12x^2 - 18x - 34
\][/tex]
4. Determine Flattening Points:
To find where the function "flattens out," we need both the first and second derivatives to be zero at the same point. Check the critical points in the second derivative:
[tex]\[
12x^2 - 18x - 34 = 0
\][/tex]
However, after evaluating, no real solutions satisfy both [tex]\( f'(x) = 0 \)[/tex] and [tex]\( f''(x) = 0 \)[/tex].
5. Conclusion:
Since we did not find any points where both the first and second derivatives equal zero, the polynomial [tex]\( f(x) \)[/tex] does not have any zeros where it "flattens out."
Thus, the number of zero(s) at which [tex]\( f \)[/tex] "flattens out" is none.
Here's a step-by-step guide to find these points:
1. Find the First Derivative:
The first derivative of the function [tex]\( f(x) \)[/tex] will help us find the critical points. Let's calculate it:
[tex]\[
f'(x) = \frac{d}{dx}(x^4 - 3x^3 - 17x^2 + 21x + 70) = 4x^3 - 9x^2 - 34x + 21
\][/tex]
2. Solve for Critical Points:
Critical points occur where the first derivative is zero, [tex]\( f'(x) = 0 \)[/tex]. Solve the equation:
[tex]\[
4x^3 - 9x^2 - 34x + 21 = 0
\][/tex]
By solving this cubic equation, we can find the critical points.
3. Find the Second Derivative:
The second derivative will help identify where the function "flattens out." Calculate the second derivative:
[tex]\[
f''(x) = \frac{d}{dx}(4x^3 - 9x^2 - 34x + 21) = 12x^2 - 18x - 34
\][/tex]
4. Determine Flattening Points:
To find where the function "flattens out," we need both the first and second derivatives to be zero at the same point. Check the critical points in the second derivative:
[tex]\[
12x^2 - 18x - 34 = 0
\][/tex]
However, after evaluating, no real solutions satisfy both [tex]\( f'(x) = 0 \)[/tex] and [tex]\( f''(x) = 0 \)[/tex].
5. Conclusion:
Since we did not find any points where both the first and second derivatives equal zero, the polynomial [tex]\( f(x) \)[/tex] does not have any zeros where it "flattens out."
Thus, the number of zero(s) at which [tex]\( f \)[/tex] "flattens out" is none.
