Question

Find f. f ″(x) = −2 + 24x − 12x2, f(0) = 2, f ′(0) = 16

Answer 1

`f''(x)=-2+24x-12x^2\hspace{5em}f'(0)=16\hspace{5em}f(0)=2 \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ \displaystyle \int~(-2+24x-12x^2)dx\implies -2x+12x^2 - 4x^3 + C=f'(x) \\\\\\ \stackrel{\textit{since we know that}}{f'(0)=16}\hspace{5em}-2(0)+12(0)^2-4(0)^3+C=16 \\\\\\ C=16\hspace{5em}\stackrel{\textit{so then we know that}}{-2x+12x^2 - 4x^3 + 16=f'(x)} \\\\[-0.35em] ~\dotfill`

`\displaystyle \int~(-2x+12x^2 - 4x^3 + 16)dx\implies -x^2+4x^3-x^4+16x+C=f(x) \\\\\\ \stackrel{\textit{since we know that}}{f(0)=2}\hspace{5em}-(0)^2+4(0)^3+16(0)+C=2 \\\\\\ C=2\hspace{5em}\boxed{-x^2+4x^3-x^4+16x+2=f(x)}`