Let $x0,y0|x^p-y^p||x-y{|}^{p}yxf(x)=x^p-y^p-(x-y{)}^{p}y{f}^{\text{'}}(x)f^{\text{'}}(x)0f(y)f(x)xyf(x)=x^{\frac{1}{7}}[0,)0.$ For $x0,y0,$ prove that $|x^p-y^p||x-y{|}^p.$ Follow the steps:

i Assume $yx.$ Let $f(x)=x^p-y^p-(x-y{)}^p.$ Fix $y$ and compute the derivative $f^{\text{'}}(x).ii$ Show that $f^{\text{'}}(x)0.$

iii Compute $f(y)$

$iv$ Deduce from $i,ii,$ and iii that $f(x)is$ non$-$positive for all $xy.$ This should prove the desired inequality.

$v$ Use the inequality $to$ show that the function $f(x)=x^{\frac{1}{7}}is$ uniformly continuous $on[0,).$

Note The method used $in$ this problem $is$ a technique that uses the derivatives $to$ prove a myriad $of$ inequalities.