Given that f(x)=k x^3-9 k x^2+9 x+3 is an increasing function on R set of real numbers, then k belongs to

Step-by-step solution

The function f(x) = kx^3 - 9kx^2 + 9x + 3 is increasing on R if and only if its derivative f'(x) 0 for all x R . Calculating the derivative: f'(x) = 3kx^2 - 18kx + 9 . For f'(x) 0 for all x , the quadratic expression 3kx^2 - 18kx + 9 must satisfy two conditions: 1. The coefficient of x^2 must be positive, so 3k > 0 , which implies k > 0 . 2. The discriminant D must be less than or equal to zero, so D 0 . Calculating the discriminant D = b^2 - 4ac : D = (-18k)^2 - 4(3k)(9) = 324k^2 - 108k . Setting D 0 : 324k^2 - 10