If f(x)=3x^5−10x^3+5, on what intervals is f′′(x)>0?

Answer 1

Sasha, the easiest way to find out would be to simply find the second derivative of your function.

f'(x) = (5)(3)x⁴ - (3)(10)x²

f''(x) = (4)(5)(3)x³-(2)(3)(10)x = 60x³-60x = 60x(x²-1)

Where is that function greater than 0? You can see pretty easily that and value of x ∈ (-1,0) will be a positive term, and then any positive x greater than 1 will allow f''(x) > 0

Therefore your answer is:

f''(x) > 0 for x ∈ (-1,0) or x > 1