Find the domain over which the curve y=x^3+12x^2 +45x -30 is increasing
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The function y=x^3+12x^2+45x-30 is increasing in the set where y' is positive.
y' = 3x^2 + 24x + 45
3x^2 + 24x + 45 > 0
=> x^2 + 8x + 15 > 0
=> x^2 + 5x + 3x + 15 > 0
=> x(x + 5) + 3(x + 5) > 0
=> (x + 3)(x + 5) > 0
This is true if:
(x + 3) > 0 and (x + 5) > 0
=> x > -3 and x > -5
=> x > -3
(x + 3) < 0 and (x + 5) < 0
=> x < -3 and x < -5
=> x < -5
The set of values of x for which the function is increasing is `(-oo, -5)U(-3, oo)`
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