Find f. f?''(x) = 20x^3 + 12x^2 + 6, f(0) = 3, f(1) = -1

Question

Find f.

f ''(x) = 20x^3 + 12x^2 + 6, f(0) = 3, f(1) = −1

Answer 1

f"(x) = 20x³+12x²+6

 

f'(x) = ∫f"(x)dx = 5x⁴+4x³+6x+C₁

 

f(x) = ∫f'(x)dx = x⁵+x⁴+3x²+C₁x+C₂

 

Since f(0) = 3, C₂ = 3

 

So, f(x) = x⁵ + x⁴ + 3x² + C₁x + 3

 

Since f(1)=-1, 8 + C₁ = -1

 

                            C₁ = -9

 

Therefore, f(x) = x⁵ + x⁴ + 3x² - 9x + 3