If x is real, the maximum value of (3x² + 9x + 17) / (3x² + 9x + 7) is (A) 1/4 (B) 41 (C) 1 (D) 17/7

Q: Why can't we just differentiate the function to find the maximum?

You absolutely can! But differentiation is like using a sledgehammer to crack a nut. It is computationally heavy and prone to errors. Algebraic manipulation is elegant, faster, and helps you understand the structure of the function better.

Q: Wait, why do we minimize the denominator to maximize the whole fraction?

Think about the structure $y = 1 + \frac{10}{g(x)}$. Since $10$ is positive, as $g(x)$ gets smaller, the fraction $\frac{10}{g(x)}$ gets larger. If $g(x)$ were negative, the logic would flip, but here $g(x)$ is always positive.

Q: How do I know the denominator is always positive?

A quadratic $ax^2 + bx + c$ with $a > 0$ is always positive if its discriminant $D = b^2 - 4ac$ is negative. Here $D = 81 - 84 = -3 < 0$. Since the parabola opens up and never touches the x-axis, it stays strictly above the x-axis.

Question

If $x$ is real, the maximum value of $\frac{3x^2 + 9x + 17}{3x^2 + 9x + 7}$ is

Options:

(A) 1/4

(B) 41

(C) 1

(D) 17/7

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler of the JEE journey! Today, we face a problem that might look intimidating at first: maximizing the rational function:

$$y = \frac{3x^2 + 9x + 17}{3x^2 + 9x + 7}$$

Many students would immediately reach for the quotient rule of differentiation, but let us pause. In the world of competitive exams, speed and elegance are your best friends.

Look closely at the numerator and the denominator. They are nearly identical! This is a classic setup for a 'surgical strike' in algebra.

We can rewrite the numerator as $(3x^2 + 9x + 7) + 10$. By splitting the fraction, we get:

$$y = 1 + \frac{10}{3x^2 + 9x + 7}$$

Now, the problem transforms. We are no longer dealing with a complex rational function; we are dealing with a simple inverse relationship.

To make $y$ as large as possible, we must make the fraction $\frac{10}{3x^2 + 9x + 7}$ as large as possible. Since the numerator $10$ is fixed, this means we must make the denominator $3x^2 + 9x + 7$ as small as possible.

### The Geometry of the Parabola

Now, let us visualize the denominator $g(x) = 3x^2 + 9x + 7$. This is a quadratic expression.

Because the leading coefficient $a = 3$ is positive, this parabola opens upwards, like a cup. It has a unique minimum point, the vertex.

We don't need calculus to find this! We can use the vertex formula:

$$g_{min} = \frac{4ac - b^2}{4a}$$

Plugging in $a = 3$, $b = 9$, and $c = 7$, we get:

$$g_{min} = \frac{4(3)(7) - (9)^2}{4(3)} = \frac{84 - 81}{12} = \frac{3}{12} = \frac{1}{4}$$

This is the smallest value the denominator can ever take.

### The Final Triumph

With the minimum denominator in hand, we return to our simplified expression:

$$y_{max} = 1 + \frac{10}{1/4}$$

Dividing by a fraction is the same as multiplying by its reciprocal, so $10 \div \frac{1}{4} = 10 	imes 4 = 40$.

Adding the constant $1$, we arrive at **$y_{max} = 41$**.

It is a clean, beautiful integer. This is the power of algebraic insight—turning a daunting problem into a simple, elegant solution. Keep this mindset, and you will conquer any problem the JEE throws at you!

Animated Solution for Mathematics - Quadratic Equations: If $x$ is real, the maximum value of $\frac{3 x ^{2} + 9 x + 17}{3 x ^{2} + 9 x + 7}$ is

Select Answer:

Visualized Solution

Analyze the Rational Function

Simplify the Numerator

Isolate the Constant Term

The Maximization Strategy

Analyze the Quadratic Denominator

The Vertex Formula

Calculate the Minimum Denominator

Substitute Back to Find $y_{ma x}$

Conclusion & Graphical Takeaway

The Sigma Insight: Maximum and Minimum Values of Quadratic Expressions

Analyzing the Setup

The Geometry of the Parabola

The Final Triumph

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Animated Solution for Mathematics - Quadratic Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If x is real, the maximum value of 3x2+9x+73x2+9x+17​ is