To determine the inflection points of the function [tex]\( f(x) = 12x^5 + 45x^4 - 80x^3 + 6 \)[/tex], follow these steps:
1. Find the second derivative of the function:
- First, calculate the first derivative [tex]\( f'(x) \)[/tex].
[tex]\[ f'(x) = \frac{d}{dx} (12x^5 + 45x^4 - 80x^3 + 6) = 60x^4 + 180x^3 - 240x^2 \][/tex]
- Next, find the second derivative [tex]\( f''(x) \)[/tex].
[tex]\[ f''(x) = \frac{d}{dx} (60x^4 + 180x^3 - 240x^2) = 240x^3 + 540x^2 - 480x \][/tex]
2. Set the second derivative equal to zero to find potential inflection points:
[tex]\[ 240x^3 + 540x^2 - 480x = 0 \][/tex]
Factor out the common term [tex]\( x \)[/tex]:
[tex]\[ x (240x^2 + 540x - 480) = 0 \][/tex]
This gives us one solution immediately:
[tex]\[ x = 0 \][/tex]
3. Solve the quadratic equation [tex]\( 240x^2 + 540x - 480 = 0 \)[/tex]:
Simplify the equation by dividing by 60:
[tex]\[ 4x^2 + 9x - 8 = 0 \][/tex]
Use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 4 \)[/tex], [tex]\( b = 9 \)[/tex], and [tex]\( c = -8 \)[/tex]:
[tex]\[ x = \frac{-9 \pm \sqrt{81 + 128}}{8} = \frac{-9 \pm \sqrt{209}}{8} \][/tex]
This yields two solutions:
[tex]\[ x = \frac{-9 + \sqrt{209}}{8} \][/tex]
[tex]\[ x = \frac{-9 - \sqrt{209}}{8} \][/tex]
4. Converting the results into numerical values and ordering them:
- The roots can be approximated numerically:
[tex]\[
\frac{-9 - \sqrt{209}}{8} \approx -2.9321
\][/tex]
[tex]\[
\frac{-9 + \sqrt{209}}{8} \approx 0.63
\][/tex]
5. Summarizing the inflection points:
- The first root delivers an inflection point approximately at [tex]\( -2.9321 \)[/tex].
- The second root provides an inflection point approximately at 0.
- The third root gives an inflection point approximately at 0.63.
Therefore, listing the inflection points from left to right, we have:
- [tex]\(x = \frac{-9 - \sqrt{209}}{8} \approx -2.93\)[/tex] (though the problem states [tex]\( -2.88 \)[/tex] which might be a rounding inconsistency to manual calculations)
- [tex]\( x = 0 \)[/tex]
- [tex]\( x = \frac{-9 + \sqrt{209}}{8} \approx 0.63 \)[/tex]
Hence, filling in the boxes and correcting the response:
- For [tex]\( D \)[/tex], we should have [tex]\( -2.9321 \)[/tex].
- For [tex]\( E \)[/tex], we have the exact [tex]\( 0 \)[/tex].
- For [tex]\( F \)[/tex], we have [tex]\( 0.63 \)[/tex].
So, the corrected and precise inflection points are:
[tex]\( D \approx -2.9321 \)[/tex], [tex]\( E = 0 \)[/tex], and [tex]\( F \approx 0.63 \)[/tex].
