Question

Let $y=f(x)(f: R \rightarrow R)$ be an explicit function defined by the implicit equation $x^3+y^3+3\left(x^2+y^2\right)+3(x+y)=14$ and $g$ be the inverse of $f$. If $\frac{d}{d x}(f(x+g(x)) \cdot g(x+f(x)))$ at $x =-1+\sqrt[3]{15}$ is equal $\lambda(15)^{2 / 3}$, where $\lambda \in I$, then find the value of $|\lambda|$.

Answer 1

$(x+1)^3+(y+1)^3=16$
$y=-1+\left(16-(x+1)^3\right)^{1 / 3}=f(x)$
$x=-1+\left(16-(y+1)^3\right)^{1 / 3} $
$\therefore f^{-1}(x)=-1+\left(16-(x+1)^3\right)^{1 / 3}=g(x) $
$\Rightarrow f(x)=g(x) $
$f(-1+\sqrt[3]{15})=g(-1+\sqrt[3]{15})=0 $
$(f(x+g(x)) \cdot g(x+f(x)))=(f(x+f(x)))^2$
$\frac{d}{d x}(f(x+f(x)))^2=2 f(x+f(x)) \cdot f^{\prime}(x+f(x)) \cdot\left(x+f^{\prime}(x)\right)$
$\frac{d}{d x}(f(x+f(x)))^2 \text { at } x=-1+\sqrt[3]{15}=0$