Question

If f' (x) = 8x³ − 2x, f(2) = 8, find f(x)

Answer 1

\[f'\left( x \right) = 8 x^3 - 2x   f\left( 2 \right) = 8\]
\[f'\left( x \right) = 8 x^3 - 2x\]
\[\int{f}'\left( x \right)dx = \int\left( 8 x^3 - 2x \right)dx\]
\[ = 8\int x^3 dx - 2\  ∫ \text{ x dx}\]
\[f\left( x \right) = 8 \left[ \frac{x^4}{4} \right] - 2 \times \frac{x^2}{2} + C\]
\[f\left( x \right) = 2 x^4 - x^2 + C\]
\[f\left( 2 \right) = 8 \left( Given \right)\]
\[f\left( 2 \right) = 2 \times 2^4 - 2^2 + C\]
\[8 = 32 - 4 + C\]
\[C = - 20\]
\[ \therefore f\left( x \right) = 2 x^4 - x^2 - 20\]