Math Problem Statement

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

Solution

Given the function f(x)=2x3+Ax2+4x1f(x) = 2x^3 + Ax^2 + 4x - 1, and knowing that f(2)=1f(2) = 1, we need to solve for AA.

First, substitute x=2x = 2 into the function:

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

Simplify the expression:

f(2)=2(8)+A(4)+4(2)1f(2) = 2(8) + A(4) + 4(2) - 1 1=16+4A+811 = 16 + 4A + 8 - 1 1=23+4A1 = 23 + 4A

Now, solve for AA:

123=4A1 - 23 = 4A 22=4A-22 = 4A A=224=5.5A = \frac{-22}{4} = -5.5

Thus, A=5.5A = -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)f(x)?
  2. What happens to the value of f(x)f(x) when x=2x = -2?
  3. How does changing AA affect the shape of the graph of f(x)f(x)?
  4. What are the critical points of f(x)f(x)?
  5. How would you solve for AA if given a different point, such as f(1)f(1)?

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

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Math Problem Analysis

Mathematical Concepts

Polynomial Functions
Substitution
Solving Equations

Formulas

Polynomial function formula
Substitution method

Theorems

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Suitable Grade Level

Grades 10-12