Let R denote the set of all real numbers. For a real number x , let [x] denote the greatest integer less than or equal to x . Let n denote a natural number. Match each entry in List-I to the correct entry in List-II and choose the correct option. List-I List-II (P) The minimum value of n for which the function f(x)= [ 10 x^3-45 x^2+60 x+35 n ] is continuous on the interval [1,2] ,is (1) 8 (Q) The minimum value of n f

Step-by-step solution

aligned & P(x)=10 x^3-45 x^2+60 x+35 & P^ (x)=30(x-1)(x-2) aligned P ( x ) decreases in [1,2] Range of P ( x )=[55,60]f( x )= [ P ( x ) n ] value of n =9 (Q) For g(x) to be increasing 2 n^2-13 n-15 0 (R) aligned & h(x)= (x^2-9 )^n (x^2+2 x+3 ) & h^ (x)= (x^2-9 )^n(2 x+2)+ (x^2+2 x+3 ) n (x^2-9 )^ n-1 2 x & = (x^2-9 )^ n-1 [2 (x^2-9 )(x+1)+2 n x (x^2+2 x+3 ) ] & =(x+3)^ n-1 (x-3)^ n-1 q(x) aligned Derivative must change sign at x=3 aligned & n -1= odd & n = even & n =6 aligned (S) | x - k + 1 2 | is differentiable e