Key Concepts and Formulas

• Range of a Rational Function: To find the range of a rational function of the form $y = \frac{ax^2+bx+c}{dx^2+ex+f}$, transform the expression into a quadratic equation in $x$.
• Discriminant: For a quadratic equation $Ax^2 + Bx + C = 0$, the discriminant is given by $D = B^2 - 4AC$.
• Real Roots and Discriminant: For real values of $x$ to exist in a quadratic equation, the discriminant must be non-negative, i.e., $D \ge 0$.

Step-by-Step Solution

Step 1: Set the Expression Equal to $y$ and Rearrange into a Quadratic Equation in $x$ Let the given expression be equal to $y$: $y = \frac{3x^2 + 9x + 17}{3x^2 + 9x + 7}$ Our goal is to find the values of $y$ for which there exist real values of $x$. To do this, we first cross-multiply to eliminate the fraction and then rearrange the terms to form a quadratic equation in $x$: $y(3x^2 + 9x + 7) = 3x^2 + 9x + 17$ $3yx^2 + 9yx + 7y = 3x^2 + 9x + 17$ Now, move all terms to one side to get a standard quadratic form $Ax^2 + Bx + C = 0$: $3yx^2 - 3x^2 + 9yx - 9x + 7y - 17 = 0$ Factor out $x^2$ and $x$ terms: $(3y - 3)x^2 + (9y - 9)x + (7y - 17) = 0$ We can further factor out common terms from the first two coefficients: $3(y - 1)x^2 + 9(y - 1)x + (7y - 17) = 0$ Here, we identify the coefficients of the quadratic equation in $x$: $A = 3(y-1)$, $B = 9(y-1)$, and $C = (7y-17)$.

Step 2: Consider the Case where the Coefficient of $x^2$ is Zero Before applying the discriminant, we must check the special case where the coefficient of $x^2$ is zero. This happens if $A = 3(y-1) = 0$, which means $y=1$. If $y=1$, substitute it back into the original rational function: $1 = \frac{3x^2 + 9x + 17}{3x^2 + 9x + 7}$ Cross-multiplying gives: $3x^2 + 9x + 7 = 3x^2 + 9x + 17$ Subtracting $3x^2 + 9x$ from both sides results in: $7 = 17$ This is a false statement, a contradiction. This implies that there is no real value of $x$ for which $y$ can be equal to $1$. Therefore, $y \ne 1$. This is an important detail to remember when interpreting the final inequality.

Step 3: Apply the Discriminant Condition for Real $x$ Since $y \ne 1$, the equation $3(y-1)x^2 + 9(y-1)x + (7y - 17) = 0$ is indeed a quadratic equation in $x$. For this quadratic equation to have real solutions for $x$, its discriminant ($D$) must be greater than or equal to zero ($D \ge 0$). The discriminant formula is $D = B^2 - 4AC$. Substitute the identified values of $A$, $B$, and $C$: $D = (9(y-1))^2 - 4 \cdot [3(y-1)] \cdot (7y-17) \ge 0$ $81(y-1)^2 - 12(y-1)(7y-17) \ge 0$

Step 4: Simplify and Solve the Inequality for $y$ Notice that $(y-1)$ is a common factor in both terms. Since we already established that $y \ne 1$, we know $(y-1) \ne 0$. We can factor out $3(y-1)$ from the inequality: $3(y-1) [27(y-1) - 4(7y-17)] \ge 0$ Now, simplify the expression inside the square brackets: $3(y-1) [27y - 27 - (28y - 68)] \ge 0$ $3(y-1) [27y - 27 - 28y + 68] \ge 0$ Combine like terms within the brackets: $3(y-1) [-y + 41] \ge 0$ Multiply by $-1/3$ and reverse the inequality sign: $(y-1)(y - 41) \le 0$ This is a quadratic inequality. The critical points (roots) where the expression equals zero are $y=1$ and $y=41$. For the product of two terms to be less than or equal to zero, one term must be non-positive and the other non-negative. Graphically, this corresponds to the portion of the parabola $f(y) = (y-1)(y-41)$ that lies below or on the y-axis. This occurs when $y$ is between the roots: $1 \le y \le 41$

Step 5: Determine the Maximum Value of $y$ From the inequality $1 \le y \le 41$, the possible values of $y$ lie in the closed interval $[1, 41]$. However, from Step 2, we conclusively showed that $y \ne 1$. Therefore, the actual range for $y$ is $1 < y \le 41$. Considering this range, the maximum value that $y$ can take is $41$.

Common Mistakes & Tips

• Don't Forget the Case $A=0$: Always check if the coefficient of $x^2$ can become zero. If it does, the original equation might reduce to a linear equation or a contradiction, which needs to be handled separately before applying the discriminant.
• Reversing Inequality Sign: A very common error is forgetting to reverse the inequality sign when multiplying or dividing both sides of an inequality by a negative number.
• Factoring Correctly: Ensure accurate factoring and simplification of the discriminant inequality to avoid errors in determining the range of $y$.

Summary

By transforming the given rational expression into a quadratic equation in $x$ and meticulously applying the condition that the discriminant must be non-negative for real solutions of $x$, we systematically derived an inequality for $y$. This inequality, coupled with a critical check for the edge case where the equation might not be quadratic (i.e., when the coefficient of $x^2$ is zero), allowed us to accurately determine the permissible range of $y$. In this specific problem, the range for $y$ was found to be $1 < y \le 41$, which unequivocally indicates that the maximum value of the expression is $41$.

Final Answer The final answer is \boxed{41}, which corresponds to option (B).