Problem 150 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If \(\mathrm{x}\) is real, the maximum value of \(\frac{3 x^{2}+9 x+17}{3 x^{2}+9 x+7}\) is [2006] (A) \(1 / 4\) (B) 41 (C) 1 (D) \(17 / 7\)

Step-by-Step Solution

Answer

The maximum value is 41.

Step 1: Analyze the expression

Let's consider the expression \( y = \frac{3x^2 + 9x + 17}{3x^2 + 9x + 7} \). We are tasked with finding the maximum value of this expression. Notice that the numerator and denominator are identical except for the constant terms 17 and 7.

Step 2: Simplify the expression

Since the terms \(3x^2 + 9x\) cancel each other out when subtracted from the numerator and denominator, we're left with focusing on the constants. The expression simplifies to \( y = 1 + \frac{10}{3x^2 + 9x + 7} \).

Step 3: Analyze the second term

The term \(\frac{10}{3x^2 + 9x + 7}\) represents a decreasing function of \(x\) since \(3x^2 + 9x + 7\) is a quadratic function opening upwards. The value of \( y \) will be maximized when \(\frac{10}{3x^2 + 9x + 7}\) is as large as possible.

Step 4: Find the minimum of the quadratic expression

To maximize \(\frac{10}{3x^2 + 9x + 7}\), we need to minimize the expression \(3x^2 + 9x + 7\). This quadratic function can be minimized using the vertex formula \(-\frac{b}{2a}\). For our quadratic, \( a = 3 \) and \( b = 9 \), so \( x = -\frac{9}{2 \cdot 3} = -\frac{3}{2} \).

Step 5: Calculate minimum value of the quadratic

Substitute \( x = -\frac{3}{2} \) into the expression: \( 3\left(-\frac{3}{2}\right)^2 + 9\left(-\frac{3}{2}\right) + 7 \). Simplifying, \( 3\left(\frac{9}{4}\right) - \frac{27}{2} + 7 = \frac{27}{4} - \frac{54}{4} + \frac{28}{4} = \frac{1}{4} \).

Step 6: Compute the maximum value of the original expression

The expression \( y \) becomes \( 1 + \frac{10}{1/4} = 1 + 40 = 41 \). Therefore, the maximum value of the initial expression is 41.

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function with a degree of 2. It can be written in the form:

• \( ax^2 + bx + c \)

where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the variable. Quadratic functions are known for their distinct U-shaped graphs called parabolas.
When \( a > 0 \), the parabola opens upwards. Conversely, if \( a < 0 \), it opens downwards. The vertex of the parabola is the point at which it changes direction.
Characteristics of a quadratic function include its vertex, axis of symmetry, and the fact that it has a maximum or minimum value depending on whether it opens upwards or downwards.

Vertex Formula

The vertex formula is a crucial part of understanding quadratic functions. It helps in finding the vertex of the parabola. For a quadratic function written in the standard form \( ax^2 + bx + c \), the vertex \( (h, k) \) is found using the vertex formula:

• \( h = -\frac{b}{2a} \)
• \( k = f(h) \) or substituting \( x = h \) into the quadratic to get \( k \)

For example, in the quadratic expression \( 3x^2 + 9x + 7 \) from the problem, the vertex can be quickly determined by:

• Setting \( a = 3 \) and \( b = 9 \)
• Calculating \( h = -\frac{9}{2 \cdot 3} = -\frac{3}{2} \)

Once \( h \) is found, substitute it back into the quadratic function to find \( k \), completing the vertex \((h, k)\), which helps in further analysis like finding the maximum or minimum value.

Maximum Value

Finding the maximum or minimum value of a quadratic function is essential in optimization problems. In this context, the maximum value is related to certain features of the expression.
For the given problem, the expression was simplified and focused on how the second term, \( \frac{10}{3x^2 + 9x + 7} \), behaves. The task was to maximize this expression by minimizing \(3x^2+9x+7\), since their relationship is inversely proportional.
Upon using the vertex formula to achieve the minimum of \(3x^2+9x+7\), it was determined that substituting this minimum value into the simplified expression was key.
The resulting maximum value of the entire original expression was found to be 41 using changes in ratio and analysis derived from quadratic behavior, showcasing how the maximum or minimum of a quadratic function impacts the entire problem.