Key Concepts and Formulas

• Range of a Rational Function: For a rational function $y = \frac{ax^2+bx+c}{dx^2+ex+f}$, the range can be found by rearranging into a quadratic in $x$ and applying the discriminant condition $D \geq 0$ for real roots.
• Discriminant of a Quadratic: For a quadratic equation $Ax^2 + Bx + C = 0$, the discriminant is $D = B^2 - 4AC$. The quadratic has real roots if and only if $D \geq 0$.
• Greatest Common Divisor (GCD): The GCD of two integers $m$ and $n$ is the largest positive integer that divides both $m$ and $n$. If $\text{gcd}(m, n) = 1$, then $m$ and $n$ are coprime.

Step-by-Step Solution

Step 1: Set up the equation $y = f(x)$ and check the denominator.

We are given the function $f(x) = \frac{2x^2 - 3x + 8}{2x^2 + 3x + 8}$. To find the range, we set $y = f(x)$: $y = \frac{2x^2 - 3x + 8}{2x^2 + 3x + 8}$

We must ensure the denominator is never zero. The discriminant of the denominator $2x^2 + 3x + 8$ is $D_{den} = 3^2 - 4(2)(8) = 9 - 64 = -55$. Since $D_{den} < 0$ and the leading coefficient is positive, the denominator is always positive and never zero.

Step 2: Rearrange the equation into a quadratic in $x$.

Multiply both sides by the denominator: $y(2x^2 + 3x + 8) = 2x^2 - 3x + 8$ $2yx^2 + 3yx + 8y = 2x^2 - 3x + 8$ Rearrange to form a quadratic equation in $x$: $(2y - 2)x^2 + (3y + 3)x + (8y - 8) = 0$

Step 3: Apply the discriminant condition for real roots.

For $x$ to be real, the discriminant of the quadratic equation must be non-negative: $D \geq 0$. $D = (3y + 3)^2 - 4(2y - 2)(8y - 8) \geq 0$

Step 4: Simplify the discriminant inequality.

Simplify the inequality: $(3(y+1))^2 - 4(2(y-1))(8(y-1)) \geq 0$ $9(y+1)^2 - 64(y-1)^2 \geq 0$ $[3(y+1) - 8(y-1)][3(y+1) + 8(y-1)] \geq 0$ $[3y + 3 - 8y + 8][3y + 3 + 8y - 8] \geq 0$ $[-5y + 11][11y - 5] \geq 0$ Multiply by $-1$ and reverse the inequality: $(5y - 11)(11y - 5) \leq 0$

Step 5: Solve the inequality for $y$.

The critical points are $y = \frac{11}{5}$ and $y = \frac{5}{11}$. Since the inequality is $(5y - 11)(11y - 5) \leq 0$, $y$ must lie between the roots: $\frac{5}{11} \leq y \leq \frac{11}{5}$

Step 6: Determine the range and the minimum and maximum values.

The range of $f(x)$ is $\left[\frac{5}{11}, \frac{11}{5}\right]$. Therefore, the minimum value is $y_{\min} = \frac{5}{11}$ and the maximum value is $y_{\max} = \frac{11}{5}$.

Step 7: Calculate the sum of the maximum and minimum values.

$y_{\min} + y_{\max} = \frac{5}{11} + \frac{11}{5} = \frac{25}{55} + \frac{121}{55} = \frac{146}{55}$

Step 8: Find $m+n$.

We have $\frac{m}{n} = \frac{146}{55}$. We need to find $\text{gcd}(146, 55)$. The prime factorization of $146$ is $2 \times 73$. The prime factorization of $55$ is $5 \times 11$. Since they share no common factors other than 1, $\text{gcd}(146, 55) = 1$. Therefore, $m = 146$ and $n = 55$. $m + n = 146 + 55 = 201$

Common Mistakes & Tips

• Sign Errors: Be extremely careful when multiplying inequalities by negative numbers. Always remember to flip the inequality sign.
• Denominator Check: Always check that the denominator is never zero before proceeding with the discriminant method.
• Coprime Condition: Always ensure that $m$ and $n$ are coprime before calculating $m+n$.

Summary

We found the range of the given rational function by setting $y$ equal to the function and rearranging into a quadratic equation in $x$. Applying the discriminant condition for real roots, we obtained an inequality for $y$, which allowed us to determine the minimum and maximum values of the function. Finally, we summed these values and found $m+n$ after ensuring $m$ and $n$ were coprime.

The final answer is \boxed{201}, which corresponds to option (C).