I am currently doing a investigation on equations called shadow functions. And I need to prove that the concavity of quintic functions are opposite. My double differentiated values of the functions are: 4(-5x^3+36x^2-69x+24) and 4(5x^3-36x^2+96x-96.

are they in opposite concavity

Thanks

A function is called concave up on an interval if it is continuous and its first derivative is continuous and increasing on the interval. If the function is twice differentiable, this is equivalent to requiring that the second derivative be positive except possibly at isolated points, where it can be zero. (Think x4, whose first derivative, 4x3, is increasing, and the second derivative is positive everywhere except at 0, where it is zero).

A function is called concave down on an interval if it is continuous and its first derivative is continuous and decreasing on the interval. If the function is twice differentiable, this is equivalent to requiring that the second derivative be negative except possibly at isolated points, where it can be zero.

in your case i dont think the first function is always negative or positive and same with the second equation