The second order derivative can be useful in maximization of a function.
This can be applied to be minimize cost function, maximize revenue function and maximize profit function.
Here, $y =5 x ^3+ x ^2+ x -10$
$\left.\therefore \frac{d y}{d x}=\frac{d}{d x}\left(5 x ^3\right)+\frac{1}{2} x ^2+\frac{3}{4} x -10\right)$
$=\frac{d}{d x}\left(5 x^3\right)+\frac{d}{d x}\left(\frac{1}{2} x ^2\right)+\frac{d}{d x}\left(\frac{3}{4} x \right)-\frac{d}{d x}(10)$
$=5 \frac{d}{d x}\left(x^3\right)+\frac{1}{2} \frac{d y}{d x}\left(x^2\right)+\frac{3}{4} \frac{d y}{d x}( x )-\frac{d}{d x}(10)$
$=5\left(3 x^2\right)+\frac{1}{2}(2 x )+\frac{3}{4}(1)-0$
$=15 x ^2+ x +\frac{3}{4}$
$\therefore \frac{d y}{d x}=15 x ^2+ x +\frac{3}{4}$
Marginal cost can be obtained by taking the derivative of cost function with respect to $x.$ Thus, when the production is $x$ then marginal cost $= \frac{dc}{dx}$.
The second order derivative of the function means the derivative of the first order derivative of the function.
It is denoted by $\frac{\mathrm{d}^2 \mathrm{y}}{\mathrm{dx}^2}$ or $f"(x).$