Answer
a: Estimated coordinates are $(\frac{3}{5},2)$
b: Actual coordinates are $(0.5824, 2.0114)$
Work Step by Step
a: With Desmos, we can see that the rightmost point on the function is somewhere around $(\frac{3}{5},2)$
b: To find the exact value, we need to find the vertical asymptote closest to the right.
$\frac{dx}{dt}=1-6t^{5}$
$\frac{dy}{dt}=e^{t}$
$\frac{dy}{dx}=\frac{e^{t}}{1-6t^{5}}$
To find the vertical asymptotes, we find when the denominator is equal to zero
$1-6t^{5}=0$
$1=6t^{5}$
$\frac{1}{5}=t^{5}$
$\sqrt[5]\frac{1}{6}=t$
Now we plug t in for x and y
$x=t-t^{6}$
$\left(\sqrt[5]{\left(\frac{1}{6}\right)}-\left(\sqrt[5]{\left(\frac{1}{6}\right)}\right)^{6}\right)= 0.5824$
$y=e^{t}$
$e^{\sqrt[5]{\left(\frac{1}{6}\right)}}=2.0114$
Thus giving us the point at $(0.5824, 2.0114)$
