Find all numbers b such that the average value of 
f(x) = 3 + 8x − 6x^2
 on the interval 
[0, b]
 is equal to 4. Enter answers as comma separated list

To find all values of \( b \) such that the average value of the function \( f(x) = 3 + 8x - 6x^2 \) on the interval \([0, b]\) is equal to 4, follow these steps:

### Step 1: Formula for the average value of a function
The formula for the average value of a function \( f(x) \) over the interval \([0, b]\) is given by:

\[
\text{Average value of } f(x) = \frac{1}{b - 0} \int_0^b f(x) \, dx
\]

### Step 2: Set up the equation
We are given that the average value is 4, so we can set up the equation as:

\[
\frac{1}{b} \int_0^b (3 + 8x - 6x^2) \, dx = 4
\]

### Step 3: Compute the integral
First, compute the integral of \( f(x) = 3 + 8x - 6x^2 \):

\[
\int (3 + 8x - 6x^2) \, dx = 3x + 4x^2 - 2x^3
\]

Now, evaluate this from 0 to \( b \):

\[
\left[ 3x + 4x^2 - 2x^3 \right]_0^b = \left( 3b + 4b^2 - 2b^3 \right) - 0 = 3b + 4b^2 - 2b^3
\]

### Step 4: Set up the equation for the average value
Substitute this result into the equation for the average value:

\[
\frac{1}{b} (3b + 4b^2 - 2b^3) = 4
\]

Simplify the equation:

\[
3 + 4b - 2b^2 = 4
\]

### Step 5: Solve the quadratic equation
Now solve the equation:

\[
3 + 4b - 2b^2 = 4
\]

Subtract 4 from both sides:

\[
-2b^2 + 4b - 1 = 0
\]

Multiply through by \(-1\) to simplify:

\[
2b^2 - 4b + 1 = 0
\]

### Step 6: Use the quadratic formula
To solve \( 2b^2 - 4b + 1 = 0 \), use the quadratic formula:

\[
b = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(2)(1)}}{2(2)}
\]

\[
b = \frac{4 \pm \sqrt{16 - 8}}{4}
\]

\[
b = \frac{4 \pm \sqrt{8}}{4}
\]

\[
b = \frac{4 \pm 2\sqrt{2}}{4}
\]

\[
b = 1 \pm \frac{\sqrt{2}}{2}
\]

### Step 7: Final answers
Thus, the solutions are:

\[
b = 1 + \frac{\sqrt{2}}{2}, \quad b = 1 - \frac{\sqrt{2}}{2}
\]

These are the two values of \( b \) where the average value of the function is 4.

Would you like more details or have any questions?

Here are 5 questions to explore:
1. What happens to the average value if the interval changes?
2. How does the quadratic formula help solve polynomial equations?
3. Can the average value method be applied to non-polynomial functions?
4. How does the function's shape affect its average over an interval?
5. How can we verify the correctness of the solutions?

Tip: The quadratic formula is a powerful tool for solving any quadratic equation!

Find all numbers b such that the average value of f(x) = 3 + 8x − 6x^2 on the interval [0, b] is equal to 4. Enter answers as comma separated list.

Solve for b such that the average value of the quadratic function f(x) = 3 + 8x − 6x^2 on the interval [0, b] equals 4. This problem involves integration and solving a quadratic equation. Suitable for advanced high school calculus students (Grades 11-12).

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