Determine Increasing/Decreasing and Concavity

Question

Given the function f(x)=4x^3+12x^2−96x, determine all intervals on which f is both increasing and concave up.

Answer 1

when f(x) is increasing, i.e, f(x)^' > 0 , take the first derivative of f(x), f(x)^' = 12x²+24x-96 > 0 , simply the equation as x²+2x-8>0 => (x+1)²> 9 => x+1 > 3 or x+1 < -3 => x> 2 or x < -4 , i.e f(x) is increasing when x>2 or x<-4 , i.e x ∈ (-∞,-4) or x∈(2,+∞)

Think about what is concave up mean, it means the slope is increasing, i.e the second derivative is positive. So take the second derivative of f(x) , f(x)^'' = 24x+24 > 0 => x>-1 , when x = -1, f(x) = 4*x³+12*x²-96x = -4+12+96=104 , f(x) is concave up in x ∈(-1,+∞).

So the interval after considering the x value for both first derivative and second derivative is x∈(2,+∞).