Find the coefficient of the squared term in the simplified form for the second derivative, f "(x) for f(x) = (x2 + 4x + 3)(4x3 − 6x2 + 8x + 1). use the hyphen symbol, -, for negative values.

Answer

120

Explanation

f(x) = (x² + 4x + 3)(4x³ − 6x² + 8x + 1)

     = (4x⁵ - 6x⁴ + 8x³ + x²) + (16x⁴ - 24x³ + 32x² + 4x) + (12x³ - 18x² + 24x + 3)

  f(x)  = 4x⁵ + 10x⁴ - 4x³ + 15x² + 28x + 3

f(x)' = 20x⁴ + 40x³ - 12x² + 30x + 28

f(x)'' = 80x³ + 120x² - 24x + 30

The coefficient of squared term is 120.

120

Step-by-step explanation:

Before we find the coefficient of the squared term, we need to expand and differentiate the resulting function twice.

Expanding the function:

f(x) = (x² + 4x + 3)(4x³ − 6x² + 8x + 1)

f(x) = 4x^5-6x^4+8x^3+x^2+16x^4-24x^3+32x^2+4x+14x^3-18x^2+24x+3

Collecting the like terms

f(x) = 4x^5+10x^4-2x^3+15x^2+28x+3

Differentiating the resulting function twice, we have;

f'(x) = 20x^4+40x^3-6x^2+30x+28

f''(x) = 80x^3 + 120x^2-12x+30

Based on the second derivative of the function, the coefficient of x² in the function is 120