D. MEAN-VALUE THEOREM & ROLLE'S THEOREMDetermine if Rolle's Theorem and the Mean Value Theorem is applicable on the specifiedinterval. If so, find a suitable value for c in that interval that satisfies the conclusion of the theorem.D. OPTIMIZATIONf(x) ={3=x²x < 1x 21on [0,1]1. A rectangle is to be inscribed in a right triangle having sides of length 6 in, 8in, and 10 in. Find the dimensions of the rectangle with greatest area assuming therectangle is positioned as the image on the right.S10 in2. A closed rectangular canister with a square base is to have a volume of 1500 cm³. It coststwice as much per square centimeter for the top and bottom as it does for the sides. Find thedimensions of the container of least cost.

Note:- Since you asked multiple question , according to our policy we can solve only first question…

D. MEAN-VALUE THEOREM & ROLLE'S THEOREM Determine if Rolle's Theorem and the Mean Value Theorem is applicable on the specified interval. If so, find a suitable value for c in that interval that satisfies the conclusion of the theorem. f(x) = {³7** x < 1 x 21 on [0,1]