The expression `x^2-4px+q^2> 0` for all real x and also `r^2+ p^2 < qr` the range of `f(x) = (x+r) / (x^2 +qx + p^2)` is

Options

A. (a)`[p/(2r),q/(2r)]`
B. (b)`(0,infty)`
C. (c)`(-infty,0)`
D. (d)`(-infty, infty)`

Answer

To solve the problem step by step, we need to analyze the given inequalities and the function \( f(x) = \frac{x+r}{x^2 + qx + p^2} \).

### Step 1: Analyze the inequality \( x^2 - 4px + q^2 > 0 \)

For the quadratic expression \( x^2 - 4px + q^2 \) to be greater than zero for all real \( x \), its discriminant must be less than zero. The discriminant \( D \) of the quadratic \( ax^2 + bx + c \) is given by \( D = b^2 - 4ac \).

Here, \( a = 1 \), \( b = -4p \), and \( c = q^2 \).

Calculating the discriminant:
\[ D = (-4p)^2 - 4 \cdot 1 \cdot q^2 = 16p^2 - 4q^2 \]

Setting the discriminant less than zero:
\[ 16p^2 - 4q^2 < 0 \]
\[ 4p^2 < q^2 \]
\[ p^2 < \frac{q^2}{4} \]

### Step 2: Analyze the inequality \( r^2 + p^2 < qr \)

This is our second condition. We keep it in mind for later use.

### Step 3: Define the function \( f(x) \)

We have:
\[ f(x) = \frac{x+r}{x^2 + qx + p^2} \]

### Step 4: Substitute \( y = x + r \)

Let \( y = x + r \), then \( x = y - r \). Substitute this into \( f(x) \):
\[ f(y) = \frac{y}{(y - r)^2 + q(y - r) + p^2} \]

Expanding the denominator:
\[ (y - r)^2 + q(y - r) + p^2 = y^2 - 2ry + r^2 + qy - qr + p^2 \]
\[ = y^2 + (q - 2r)y + (r^2 - qr + p^2) \]

### Step 5: Set up the quadratic in \( y \)

We can rewrite \( f(y) \) as:
\[ f(y) = \frac{y}{y^2 + (q - 2r)y + (r^2 - qr + p^2)} \]

### Step 6: Analyze the range of \( f(y) \)

To find the range of \( f(y) \), we need to analyze the behavior of the function. The denominator must not be zero, and we need to ensure that it is positive for the function to be defined.

The quadratic in the denominator must have a positive leading coefficient and a negative discriminant to ensure it does not cross the x-axis.

### Step 7: Conditions for the denominator

The leading coefficient is 1 (positive), and we need:
\[ (q - 2r)^2 - 4 \cdot 1 \cdot (r^2 - qr + p^2) < 0 \]

### Step 8: Combine the conditions

From the previous steps, we have:
1. \( p^2 < \frac{q^2}{4} \)
2. \( r^2 + p^2 < qr \)

These conditions will help us determine the range of \( f(y) \).

### Conclusion

Given the conditions, we analyze the behavior of \( f(y) \) as \( y \) approaches infinity and negative infinity. The function will approach 0 as \( y \) goes to either extreme, and we need to find the maximum value it can take, which will depend on the specific values of \( p, q, r \) satisfying the inequalities.