For finding the interval in which given function is increasing or decreasing ,there is no need to calculate maxima or minima.
Just differentiate the function and find roots of f'(x), and then check where sign of f(x) is positive or negative.
here f'(x) = x^2 + x -6 = 0 => x = -3 , 2
for x ∈ (-∞,-3] ∪ [2 ,∞) , f'(x) >= 0 ,hence slope of function is increasing in this interval.Hence f(x) is increasing.
I hope you can find unnecessary steps in your solution.