Question

Let $y=f(x)(f:R\to R)$ 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 $\frac{d}{dx}(f(x+g(x))\cdot g(x+f(x)))$ at $x=-1+\sqrt[3]{15}$ is equal $\lambda (15{)}^{2/3}$, where $\lambda \in I$, then find the value of $|\lambda |$.

Answer 1

Answer: 0

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