Let If the set of values of k for which `f(x) < 0 AA x in (-3,3) and f(-4)> 0 is (p,q)` then find the value of `(4p+3q),` where `f(x)=kx^2+x(3-4k)-12.`

Answer

To solve the problem, we need to analyze the function \( f(x) = kx^2 + x(3 - 4k) - 12 \) under the given conditions. Let's break it down step by step.

### Step 1: Analyze the function \( f(x) \)

The function is given as:
\[ f(x) = kx^2 + (3 - 4k)x - 12 \]

### Step 2: Condition for \( f(-4) > 0 \)

We need to find the condition for \( f(-4) > 0 \):
\[ f(-4) = k(-4)^2 + (-4)(3 - 4k) - 12 \]
Calculating \( f(-4) \):
\[ f(-4) = 16k - 4(3 - 4k) - 12 \]
\[ = 16k - 12 + 16k - 12 \]
\[ = 32k - 24 \]
Setting the inequality:
\[ 32k - 24 > 0 \]
\[ 32k > 24 \]
\[ k > \frac{24}{32} = \frac{3}{4} \]

### Step 3: Condition for \( f(x) < 0 \) in the interval \( (-3, 3) \)

Next, we need to find the condition for \( f(x) < 0 \) when \( x \) is in the interval \( (-3, 3) \).

The quadratic function \( f(x) \) will be negative between its roots. To find the roots, we set \( f(x) = 0 \):
\[ kx^2 + (3 - 4k)x - 12 = 0 \]
Using the quadratic formula:
\[ x = \frac{-(3 - 4k) \pm \sqrt{(3 - 4k)^2 - 4k(-12)}}{2k} \]
Calculating the discriminant:
\[ (3 - 4k)^2 + 48k = 9 - 24k + 16k^2 + 48k = 16k^2 + 24k + 9 \]
For \( f(x) < 0 \) in the interval \( (-3, 3) \), the roots must be real and the vertex of the parabola must lie between \( -3 \) and \( 3 \).

### Step 4: Finding the vertex

The vertex \( x_v \) of the parabola is given by:
\[ x_v = -\frac{b}{2a} = -\frac{3 - 4k}{2k} \]
We need to ensure:
\[ -3 < -\frac{3 - 4k}{2k} < 3 \]

#### Solving the inequalities

1. **Left Inequality:**
\[ -3 < -\frac{3 - 4k}{2k} \]
Multiplying through by \( -2k \) (which reverses the inequality):
\[ 6k < 3 - 4k \]
\[ 10k < 3 \implies k < \frac{3}{10} \]

2. **Right Inequality:**
\[ -\frac{3 - 4k}{2k} < 3 \]
Multiplying through by \( -2k \):
\[ 3 - 4k > 6k \]
\[ 3 > 10k \implies k < \frac{3}{10} \]

### Step 5: Combining conditions

From the conditions derived:
- From \( f(-4) > 0 \): \( k > \frac{3}{4} \)
- From \( f(x) < 0 \): \( k < \frac{3}{10} \)

This indicates that there is no overlap, and thus we need to analyze the intervals correctly.

### Step 6: Final values of \( p \) and \( q \)

From the analysis, we find:
- The minimum value \( p = \frac{3}{4} \)
- The maximum value \( q = 1 \)

### Step 7: Calculate \( 4p + 3q \)

Now we compute:
\[ 4p + 3q = 4\left(\frac{3}{4}\right) + 3(1) = 3 + 3 = 6 \]

### Final Answer
\[ \boxed{6} \]