QUESTION 3 a) The volume of an open cylindrical tank is 100m . Given that / is the radius of the circular base and h is the height of the tank. i) Show that the total surface area of the tank is given by A = z/2 + 200 ii) Find the value of r and h that will minimize the total surface area of the tank. (8 marks) b) Consider a function f (x) = x3+ 4 -2x +5. i) Find the interval(s) where the function f (x) is increasing or decreasing. ii) Find the interval(s) where the function f (x) is concave upward or downward. (8 marks) C) The Mean-Value Theorem for differentiation states that if f (x) is differentiable on (a,b) and continuous on [a, b], then there is at least one point cin (a,b) where f'(c) = _ f(b)-f(a) b- a If f(x)=4+ 2 X , find the exact value of c in the interval (1,2) that satisfies the above theorem. (4 marks)APPENDIX 1 RULES OF DIFFERENTIATION 1. Product rule (f ( x ) 9 ( x ) ) = g(x )' ( x ) + f(x )g'(x ) 2. Quotient rule d (f (x) _ g(x) f(x)- f(x)g'(x) dx g(x) (g(x)) 3. Power rule "( f( x ) ) " = n[f( x ) ]" ' f ( x ) 4. Chain rule dx "f (g(x)) = f(9(x)) g'(x) TABLE OF INTEGRALS (ax + b)7+1) +C; n#-1 1 . [( ax + b ) " dx = a(n + 1) - Inlax + b/ + C ; n=1 2 . - dx = In/ x | + c 3 . sin axdx = - - cos ax + C 4. cos axdx = - sin ax + C 5 . sec? axdx = = tan ax + C 6. csc2 axdx = --cot ax + c a