To evaluate the expression \(10x - 3 - \left( \frac{1}{4} x^2 (12 - x) \right)\) at the specified values of \(x\), we will first simplify the expression:
\[ y = 10x - 3 - \left( \frac{1}{4} x^2 (12 - x) \right) \]
Now, we can expand the term \(\frac{1}{4} x^2 (12 - x)\):
\[ \frac{1}{4} x^2 (12 - x) = \frac{1}{4} (12x^2 - x^3) = 3x^2 - \frac{1}{4} x^3 \]
So we can rewrite the expression as:
\[ y = 10x - 3 - \left( 3x^2 - \frac{1}{4} x^3 \right) \] \[ y = 10x - 3 - 3x^2 + \frac{1}{4} x^3 \] \[ y = \frac{1}{4} x^3 - 3x^2 + 10x - 3 \]
Next, we will evaluate \(y\) at \(x = 1\), \(x = 2\), \(x = 4\), and \(x = 6\):
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For \(x = 1\): \[ y(1) = \frac{1}{4}(1^3) - 3(1^2) + 10(1) - 3 \] \[ = \frac{1}{4} - 3 + 10 - 3 = \frac{1}{4} + 4 = \frac{17}{4} = 4.25 \]
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For \(x = 2\): \[ y(2) = \frac{1}{4}(2^3) - 3(2^2) + 10(2) - 3 \] \[ = \frac{1}{4}(8) - 3(4) + 20 - 3 \] \[ = 2 - 12 + 20 - 3 = 7 \]
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For \(x = 4\): \[ y(4) = \frac{1}{4}(4^3) - 3(4^2) + 10(4) - 3 \] \[ = \frac{1}{4}(64) - 3(16) + 40 - 3 \] \[ = 16 - 48 + 40 - 3 = 5 \]
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For \(x = 6\): \[ y(6) = \frac{1}{4}(6^3) - 3(6^2) + 10(6) - 3 \] \[ = \frac{1}{4}(216) - 3(36) + 60 - 3 \] \[ = 54 - 108 + 60 - 3 = 3 \]
Now we have the evaluated values:
- \(y(1) = 4.25\)
- \(y(2) = 7\)
- \(y(4) = 5\)
- \(y(6) = 3\)
Finally, let's list these from greatest to least:
- \(7\) (for \(x = 2\))
- \(5\) (for \(x = 4\))
- \(4.25\) (for \(x = 1\))
- \(3\) (for \(x = 6\))
Arranged from greatest to least: 7, 5, 4.25, 3.
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