Mark the correct alternative in the following:
If the function f(x) = x3 – 9k x2 + 27x + 30 is increasing on R, then
Formula:- (i) ax2+bx+c>0 for all x a>0 and b2-4ac<0
(ii) ax2+bx+c<0 for all x a<0 and b2-4ac<0
(iii) The necessary and sufficient condition for differentiable function defined on (a,b) to be strictly increasing on (a, b) is that f’(x)>0 for all x(a,b)
Given:-
f(x) = x3 – 9k x2 + 27x + 30
f’(x)=3x2-18kx+27
for increasing function f’(x)>0
3x2-18kx+27>0
x2-6kx+9>0
Using formula (i)
36k2-36>0
K2>1
Therefore –1 <k < 1