Question

Determine a função y = f(x) sabendo que f'''(x) = 6; f''(0) = -8; f'(0) = 0 e f(0) = 5

Answer 1

$\int {f'''(x)} \, dx =\int 6dx\\f''(x) = 6x + c_1\\f''(0)=6*0+c_1 = -8\\c_1=-8\\f''(x) = 6x-8\\\\\int {f''(x)}dx=\int{(6x-8)}dx\\f'(x)=3x^2-8x+c_2\\f'(0)=3*0^2-8*0+c_2 = 0\\c_2=0\\f'(x)=3x^2-8x\\\\\int{f'(x)}dx=\int{(3x^2-8x)}dx\\f(x)=x^3-4x^2+c_3\\f(0)=0^3-4*0^2+c_3=5\\c_3=5\\f(x)=x^3-4x^2+5$