Word Count : 349

---

We are to use the Implicit Function Theorem to prove the existence of a unique differentiable function $g(y)$ such that $$ g(1) = 2 \quad \text{and} \quad F(g(y),y) = 0 $$ in a neighborhood of $(x_0,y_0) = (2,1)$, where $$ F(x,y) = x^5 + y^5 - 16xy^3 - 1. $$ Also, we need to compute $g'(y)$. --- ### Step 1: Verify $F(2,1)=0$ Check that $(x,y) = (2,1)$ satisfies $F(x,y)=0$: __________ ______ _________ _____ _____ ________ ______ _______ _____ ________ _______.
______ _______ ________ _____ _________ ____ ___.
________ _________ ______ ____ ______.
___ __________ _______ ____ ____ _______ __________ __________ ____ ________.
___ ____ ________ _____ _____ ________ ___ _______ ____ ______.
_____ ______ _______ _________ _____ _____ ______ __________ ____ _________ _________ _____.
________ __________ ________ _________ ____ _____ ____ ______.
_____ _____ _____ __________ _____ ____ ________ ___ ________.
_______ ______ ___ _____ __________ _____ ___.
_________ ____ ____ ________ ______ _______ __________ _____ _______ ______.
___ ______ ________ ______ __________ ________ _____ ______.
_________ _______ ______ __________ _______ __________ _____ __________ ______ _______.
________ __________ _______ _________ ________ ______ _____ _____ ________.
___ _____ __________ ____ ________ ___.
__________ ____ ______ ___ _________ ______.
_______ ______ _______ _________ _________ _______ __________ _______ _____ _______.
_________ _________ ____ ______ _______ __________.
______ _________ ______ ________ _______ __________ ________ _______ ____ __________.
________ ___ __________ __________ ____ _____.
____ ______ ___ ________ _________ ______ __________ ______ _________ ______ _____ _____.
________ ___ _______ ___ ________ ___ _________ ______ _____ __________ _____ _______.
___ ________ _______ _______ _____ _______ _________ ___ ________ ____.
_______ ___ ___ _________ _________ ______ __________ ____ ____ ______ _________.
________ _______ __________ __________ _____ __________ ______ _______ ____ _________.
____ ____ __________ _________ __________ _____ _________ ________ __________ ______.
_______ ______ ____ _______ _____ ________ _____.
___ ____ __________ _____ _______.
_________ ________ _______ ___ _________ ___ ____.
____ ________ _________ ____ ________ _______ ____ _______ ________ _________ __________.
________ ________ __________ _______ ___ ______.
___ ____ _________ ______ _______ ____ _______ _______ _______ ___.
__________ ______ ___ _____ ______ __________ _______ ____ _____.
Get Full Answer on WhatsApp