Could I get help for these please?

Question 8 ? (a) The numbers p, q and s are fixed and positive. Also p > 1, q > 1 and P= q q- 1 i. What positive value of t minimises the expression 2 f (t) = + p - - st? ii. Show that for all t > 0, SP p 2 st iii. Prove by induction that 4 (172 X . . . X In) S n\fQuestion 8. (a) Consider the degree n version of the AM/GM inequality which states for positive real numbers x 1, X2, ..., *n that Xitx2 + " + Xn n 2 1 X1X2 ." Xn (i) Prove that if the AM/GM inequality is true for n = k, then it will be true for n = 2k (ii) Prove that if the AM/GM inequality is true for n = k, then it will be true for n = k - 1Question 8 b (3} Suppose that a and b axe DOGitive real numbers, and let at) = 03: E:- :5- a :t a a+b 2 i. Show that the minimum value of x) occurs at 3: = ii. Suppose that c is. a positive real number. a+b+c 3 e+b 2 a+b+c Show that (SSM) 2 (2m) and deducer213 etc. b You may assume that % .2 m. iii. Suppose that the cubic equation 3:3 391:2 + grit r = 0 has three positive real roots. Use part (ii) to prove that p3 2 27:". iv. Deduco that the cubic equation 3:3 " 2:52 + a: 1 = 0 has exactly one real root