if f(x)=2x^(3)+Ax^(2)+4x-1;f(2)=1, solve for A

Given the function \( f(x) = 2x^3 + Ax^2 + 4x - 1 \), and knowing that \( f(2) = 1 \), we need to solve for \( A \).

First, substitute \( x = 2 \) into the function:

\[
f(2) = 2(2)^3 + A(2)^2 + 4(2) - 1
\]

Simplify the expression:

\[
f(2) = 2(8) + A(4) + 4(2) - 1
\]
\[
1 = 16 + 4A + 8 - 1
\]
\[
1 = 23 + 4A
\]

Now, solve for \( A \):

\[
1 - 23 = 4A
\]
\[
-22 = 4A
\]
\[
A = \frac{-22}{4} = -5.5
\]

Thus, \( A = -5.5 \).

Would you like any further details or explanations?

Here are five related questions:
1. How would you find the derivative of \( f(x) \)?
2. What happens to the value of \( f(x) \) when \( x = -2 \)?
3. How does changing \( A \) affect the shape of the graph of \( f(x) \)?
4. What are the critical points of \( f(x) \)?
5. How would you solve for \( A \) if given a different point, such as \( f(1) \)?

**Tip:** When solving for constants, always carefully substitute values and simplify before attempting to isolate the variable.

Learn how to solve for the constant A in the polynomial function f(x) = 2x^3 + Ax^2 + 4x - 1, given f(2) = 1. This problem involves polynomial functions and substitution methods. Suitable for high school students.

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