Question: Using the principle of mean-value theorem on the indicated interval, find all numbers c that satisfy the conclusion of the theorem.?

(a) \(f\left( x \right)=x+{{\left( x-3 \right)}^{2}},\,\,\left[ 1,4 \right]\)

(b) \(f\left( x \right)=5{{\left( 1+2x \right)}^{1/2}},\,\,\,\left[ 0,4 \right]\)