Understanding the Problem

We are given a function f: R → R defined as:

f(x) = x³ + x² f'(1) + x f''(2) + f'''(3)

Our goal is to find f(2). Notice that f'(1), f''(2), and f'''(3) are constants because they are derivatives evaluated at specific points. Let's denote:

Let a = f'(1), b = f''(2), c = f'''(3)

So, f(x) = x³ + a x² + b x + c

Step 1: Find the First Derivative f'(x)

Differentiate f(x) with respect to x:

Step 2: Find the Second Derivative f''(x)

Differentiate f'(x) with respect to x:

$f\text{'}\text{'}\left(x\right)=\frac{d}{dx}(3x^2+2ax+b)=6x+2a$

Step 3: Find the Third Derivative f'''(x)

Differentiate f''(x) with respect to x:

$f\text{'}\text{'}\text{'}\left(x\right)=\frac{d}{dx}(6x+2a)=6$

This is a constant value, independent of x.

Step 4: Evaluate the Constants at the Given Points

We defined our constants as:

a = f'(1)

b = f''(2)

c = f'''(3)

From Step 3, we found that f'''(x) = 6. Therefore:

c = f'''(3) = 6

Now, let's find an expression for b using the second derivative:

f''(x) = 6x + 2a

Therefore, b = f''(2) = 6*(2) + 2a = 12 + 2a

Now, let's find an expression for a using the first derivative:

f'(x) = 3x² + 2a x + b

Therefore, a = f'(1) = 3*(1)² + 2a*(1) + b = 3 + 2a + b

This gives us an equation: a = 3 + 2a + b

Rearranging terms: a - 2a = 3 + b

-a = 3 + b

a = -3 - b ...(Equation 1)

We also have an expression for b from earlier: b = 12 + 2a ...(Equation 2)

Step 5: Solve the System of Equations

We have two equations:

1) a = -3 - b

2) b = 12 + 2a

Substitute Equation 1 into Equation 2:

b = 12 + 2(-3 - b)

b = 12 - 6 - 2b

b = 6 - 2b

b + 2b = 6

3b = 6

b = 2

Now substitute b = 2 back into Equation 1:

a = -3 - (2)

a = -5

We already found c = 6.

Step 6: Write the Complete Function f(x)

We now know all constants:

a = -5, b = 2, c = 6

Therefore, the function is:

f(x) = x³ + (-5)x² + (2)x + (6) = x³ - 5x² + 2x + 6

Step 7: Find f(2)

Substitute x = 2 into the function:

f(2) = (2)³ - 5*(2)² + 2*(2) + 6

f(2) = 8 - 5*4 + 4 + 6

f(2) = 8 - 20 + 4 + 6

f(2) = (8 + 4 + 6) - 20

f(2) = 18 - 20

f(2) = -2

Final Answer

f(2) = -2

Related Topics

• Differentiation: The process of finding the derivative of a function, which represents its instantaneous rate of change.
• Higher-Order Derivatives: Derivatives of derivatives, such as the second derivative (acceleration) or third derivative (jerk).
• Polynomial Functions: Functions of the form f(x) = a_nxⁿ + ... + a₁x + a₀, which are smooth and continuous everywhere.
• Systems of Equations: A set of equations with multiple variables that are solved simultaneously to find a common solution.

Key Formulae & Theory

Power Rule for Differentiation:

$\frac{d}{dx}\left(x^n\right)=n{x}^{n-1}$

Derivative of a Constant:

$\frac{d}{dx}\left(c\right)=0$

Linearity of Differentiation:

$\frac{d}{dx}\left[af\right(x)+bg(x\left)\right]=af\text{'}\left(x\right)+bg\text{'}\left(x\right)$

Core Concept: The key insight in this problem was recognizing that f'(1), f''(2), and f'''(3) are simply constant numbers. This allowed us to treat the original equation as a standard polynomial and use differentiation to create a solvable system of equations for these constants.