Mark the correct alternative in the following:

If the function f(x) = x^{3} – 9k x^{2} + 27x + 30 is increasing on R, then

Formula:- (i) ax^{2}+bx+c>0 for all x a>0 and b^{2}-4ac<0

(ii) ax^{2}+bx+c<0 for all x a<0 and b^{2}-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) = x^{3} – 9k x^{2} + 27x + 30

f’(x)=3x^{2}-18kx+27

for increasing function f’(x)>0

3x^{2}-18kx+27>0

x^{2}-6kx+9>0

Using formula (i)

36k^{2}-36>0

K^{2}>1

Therefore –1 <k < 1

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