The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the point on the curve farthest from the line $x = y$ [closed]

(NBHM_2006_PhD Screening Test_Analysis)

The curve $x^3− y^3= 1$ is asymptotic to the line $x = y$. Find the point on the curve farthest from the line $x = y$

how should i solve this problem

Hint: can you see why the slope of the curve at that point would be the sane as the slope of the line?

Hint: Step 1) Consider lines of the form $y = x+ C$.

$\frac {C}{\sqrt{2}}$ represents the distance between any point on the line, and the line $y=x$. We are looking for the largest absolute value of C such that a point of $x^3 - y^3 = 1 $ lies on the line. This is equivalent to solving for $\frac {dy}{dx} = A$ (some value for you to determine).

Step 2) Proceed by implicit differentiation, to get that the point satisfying $\frac {dy}{dx}=A$ also satisfies $y^2 = x^2$. Hence, solve for the point.