SOLUTION: Find the maximum value of f(a), where a is a positive integer, and
f(x) = \frac{13 + x}{2x + 7 - x^2}.
Algebra.Com
Question 1209895: Find the maximum value of f(a), where a is a positive integer, and
f(x) = \frac{13 + x}{2x + 7 - x^2}.
Found 2 solutions by CPhill, greenestamps:
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Let $f(x) = \frac{13+x}{2x+7-x^2}$. We want to find the maximum value of $f(a)$, where $a$ is a positive integer.
First, let's analyze the denominator: $2x+7-x^2 = -(x^2-2x-7) = -(x^2-2x+1-8) = -(x-1)^2+8 = 8-(x-1)^2$.
The denominator is maximized when $x=1$, and the maximum value is 8.
We need the denominator to be positive, so $8-(x-1)^2 > 0$.
$(x-1)^2 < 8$.
$-\sqrt{8} < x-1 < \sqrt{8}$.
$1-\sqrt{8} < x < 1+\sqrt{8}$.
Since $\sqrt{8} \approx 2.828$, we have $1-2.828 < x < 1+2.828$, so $-1.828 < x < 3.828$.
Since $a$ is a positive integer, $a$ can be 1, 2, or 3.
Now let's evaluate $f(a)$ for $a=1, 2, 3$.
* $f(1) = \frac{13+1}{2(1)+7-1^2} = \frac{14}{2+7-1} = \frac{14}{8} = \frac{7}{4} = 1.75$
* $f(2) = \frac{13+2}{2(2)+7-2^2} = \frac{15}{4+7-4} = \frac{15}{7} \approx 2.143$
* $f(3) = \frac{13+3}{2(3)+7-3^2} = \frac{16}{6+7-9} = \frac{16}{4} = 4$
The maximum value of $f(a)$ is 4, which occurs when $a=3$.
**Therefore, the maximum value of f(a) is 4.**
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!
The critical points are where the numerator or denominator is zero.
The numerator is 0 when x = -13.
The zeros of the denominator are and . Those values are approximately -1.8 and +3.8.
We are looking for the maximum value of the function for positive integer x values.
When x is very large positive, the denominator is positive and the numerator is negative, so the function value is negative. Then since the critical points are at -13, about -1.8 and about +3.8, the function value is positive only between the two zeros of the denominator.
Since we are looking for the maximum value of the function for positive integer values of x, the answer has to be at x equal to 1, 2, or 3.
ANSWER: 4
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