$r^{th}$ rootsrin$2,3,4,$dots and $C>0.$ Let $g(x)=x^r-C.$ Let $x_1>0.$

$(a)$ Find the equation $of$ the tangent line $to$ the graph $ofgat$ the point $(x_1,g(x_1)).$

$(b)$ Find the value $x_2$ where the tangent line $in$ part $(a)$ crosses the $x-$axis. Find a function $f$ such that

$x_2=f(x_1).$

$(c)$ Show that for $x>0,f(x)=xif$ and only $ifx=C^{\frac{1}{r}},$ for $fasin$ part $(b).$

$(d)$ Show that $f(x)C^{\frac{1}{r}},$ for any $x>0.$ Hint: use calculus for the required inequality.

$(e)$ Show that $f(x)xifx{C}^{\frac{1}{r}}.$