Let `f(x)=(mx^2+3x+4)/(x^(2)+3x+4) , m in R` . If `f(x) lt 5` for all ` x in R ` then the possible set of values of m is :

Options

A. `m lt (11)/(4)`
B. `m lt 5`
C. `(11)/(4) lt m lt 5`
D. None of these

Answer

To solve the problem, we need to find the values of \( m \) such that the function

\[ f(x) = \frac{mx^2 + 3x + 4}{x^2 + 3x + 4} \]

is less than 5 for all \( x \in \mathbb{R} \).

### Step 1: Set up the inequality

We start by setting up the inequality:

\[ f(x) < 5 \]

This can be rewritten as:

\[ \frac{mx^2 + 3x + 4}{x^2 + 3x + 4} < 5 \]

### Step 2: Rearranging the inequality

We can rearrange this inequality by subtracting 5 from both sides:

\[ \frac{mx^2 + 3x + 4}{x^2 + 3x + 4} - 5 < 0 \]

### Step 3: Combine into a single fraction

To combine the terms into a single fraction, we need a common denominator:

\[ \frac{mx^2 + 3x + 4 - 5(x^2 + 3x + 4)}{x^2 + 3x + 4} < 0 \]

### Step 4: Simplify the numerator

Now, we simplify the numerator:

\[ mx^2 + 3x + 4 - 5x^2 - 15x - 20 = (m - 5)x^2 + (3 - 15)x + (4 - 20) \]

This simplifies to:

\[ (m - 5)x^2 - 12x - 16 \]

Thus, we have:

\[ \frac{(m - 5)x^2 - 12x - 16}{x^2 + 3x + 4} < 0 \]

### Step 5: Analyze the denominator

Next, we need to analyze the denominator \( x^2 + 3x + 4 \). We calculate its discriminant:

\[ D = b^2 - 4ac = 3^2 - 4 \cdot 1 \cdot 4 = 9 - 16 = -7 \]

Since the discriminant is negative, the quadratic \( x^2 + 3x + 4 \) has no real roots and is always positive for all \( x \).

### Step 6: Focus on the numerator

For the entire fraction to be negative, we need the numerator to be less than zero for all \( x \):

\[ (m - 5)x^2 - 12x - 16 < 0 \]

### Step 7: Condition for the numerator

For the quadratic \( (m - 5)x^2 - 12x - 16 \) to be negative for all \( x \), it must open downwards (i.e., \( m - 5 < 0 \)) and have no real roots. This means:

1. \( m - 5 < 0 \) implies \( m < 5 \)
2. The discriminant of the quadratic must be less than or equal to zero:

\[ (-12)^2 - 4(m - 5)(-16) < 0 \]

Calculating the discriminant:

\[ 144 + 64(m - 5) < 0 \]

This simplifies to:

\[ 144 + 64m - 320 < 0 \]

\[ 64m - 176 < 0 \]

\[ 64m < 176 \]

\[ m < \frac{176}{64} = \frac{11}{4} \]

### Conclusion

Combining both conditions, we find:

\[ m < \frac{11}{4} \]

Thus, the possible set of values for \( m \) is:

\[ m < \frac{11}{4} \]