Question

Help me answer this question please​

Answer 1

f''(x) ≈ 7953.87

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

1. Brackets
2. Parenthesis
3. Exponents
4. Multiplication
5. Division
6. Addition
7. Subtraction

• Left to Right

Algebra I

• Factoring

Calculus

Derivatives

Basic Power Rule:

• f(x) = cxⁿ
• f’(x) = c·nxⁿ⁻¹

Product Rule:
`\displaystyle (d)/(dx) [f(x)g(x)]=f'(x)g(x) + g'(x)f(x)`

Derivative Rule:
`\displaystyle (d)/(dx) [e^x]=e^x`

Explanation:

Step 1: Define

f(x) = 4x³eˣ

f''(x) is x = 3 for 2nd Derivative

Step 2: Differentiate

1. [1st Derivative] Product Rule [Basic Power Rule]: f'(x) = 3 · 4x³⁻¹eˣ + 4x³eˣ
2. [1st Derivative] Simplify: f'(x) = 12x²eˣ + 4x³eˣ
3. [1st Derivative] Factor: f'(x) = eˣ(12x² + 4x³)
4. [2nd Derivative] Product Rule [Basic Power Rule]: f''(x) = eˣ(12x² + 4x³) + eˣ(2 · 12x²⁻¹ + 3 · 4x³⁻¹)
5. [2nd Derivative] Simplify: f''(x) = eˣ(12x² + 4x³) + eˣ(24x + 12x²)
6. [2nd Derivative] Distribute eˣ: f''(x) = 12eˣx² + 4eˣx³ + 24eˣx + 12eˣx²
7. [2nd Derivative] Combine like terms: f''(x) = 24eˣx² + 4eˣx³ + 24eˣx
8. [2nd Derivative] Factor: f''(x) = 4xeˣ(x² + 6x + 6)

Step 3: Evaluate

1. Substitute: f''(x) = 4(3)e³(3² + 6(3) + 6)
2. Exponents: f''(x) = 12e³(9 + 6(3) + 6)
3. Multiply: f''(x) = 12e³(9 + 18 + 6)
4. Add: f''(x) = 12e³(27 + 6)
5. Add: f''(x) = 12e³(33)
6. Multiply: f''(x) = 396e³
7. Evaluate: f''(x) ≈ 396(20.0855)
8. Multiply: f''(x) ≈ 7953.87

Answer 2

9514 1404 393

Answer:

7953.873

Explanation:

The first derivative is ...

f'(x) = 4·3x²·e^x +4x³·e^x = e^x(4x³ +12x²)

Then the second derivative is ...

f''(x) = (12x² +24x)e^x +(4x³ +12x²)e^x

f''(x) = e^x(4x³ +24x² +24x)

So, f''(3) = (e^3)(4·27 +24·9 +24·3) = 396e^3 = 7953.87262158

Rounded to thousandths, this is ...

f''(3) = 7953.873