To find where the function is concave up, we need to find the second derivative of the function, as a function is concave upward wherever its second derivative is positive.
The function given is f(t) = t⁴ - 16t³ - 9.
We start by finding the first derivative of the function, f'(t). However, as we are focused on finding where the original function is concave up, it is most important to find the second derivative f''(t).
Once we find the second derivative, we equate f''(t) to zero and solve for t to find potential points of inflection. This is because the concavity may change at these points. However, it is not guaranteed - a function can have points where the second derivative is zero, but the concavity doesn't change.
Now, rather than simply finding where the second derivative equals zero, we want to find when the second derivative is greater than zero. This is because we're looking for intervals where the function is concave up. Our goal is to solve f''(t) > 0 for t.
The solution to this inequality gives the intervals for the values of t, for which the function is concave upwards.
According to the calculations, the function is concave up when t is less than 0 and when t is more than 8.
So, the function f(t) = t⁴ -16t³ -9 is concave upward on the intervals (-∞,0) and (8, ∞).