If f(x) = 2x3 + Ax2 + 4x - 3 and f (2) = 11,
what is the value of A?

To find the value of \(A\) in the function \(f(x) = 2x^3 + Ax^2 + 4x - 3\) given that \(f(2) = 11\), we substitute \(x = 2\) into the function and set it equal to 11. This gives us:

\[
f(2) = 2(2)^3 + A(2)^2 + 4(2) - 3 = 11
\]

Now, let's simplify step-by-step:

1. Calculate \(2(2)^3\):
   \[
   2 \times 8 = 16
   \]

2. Calculate \(A(2)^2\):
   \[
   A \times 4 = 4A
   \]

3. Calculate \(4(2)\):
   \[
   4 \times 2 = 8
   \]

4. Substitute these values into the equation:
   \[
   16 + 4A + 8 - 3 = 11
   \]

5. Combine like terms:
   \[
   21 + 4A = 11
   \]

6. Isolate \(4A\) by subtracting 21 from both sides:
   \[
   4A = 11 - 21
   \]

   \[
   4A = -10
   \]

7. Solve for \(A\) by dividing both sides by 4:
   \[
   A = -\frac{10}{4} = -\frac{5}{2}
   \]

So, the value of \(A\) is:

\[
\boxed{-\frac{5}{2}}
\]

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How would the function \(f(x)\) change if a different value for \(f(2)\) were given?
2. What are the roots of the function \(f(x) = 2x^3 + Ax^2 + 4x - 3\) after finding \(A\)?
3. How would you determine the derivative \(f'(x)\) of this function?
4. Can we determine the local maxima or minima of \(f(x)\) given this information?
5. What happens to the function's graph if \(A\) were a positive number instead?

**Tip:** Always check your work by substituting your solution back into the original equation to verify the correctness.

If f(x) = 2x^3 + Ax^2 + 4x - 3 and f(2) = 11, what is the value of A?

Find the value of A in the polynomial function f(x) = 2x^3 + Ax^2 + 4x - 3 when f(2) = 11. Follow a step-by-step solution that involves algebraic substitution and simplification. This exercise is suitable for high school students, particularly those in Grades 9-12.

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