Find the value of k. P(t)= -0.02t^2 - 1.5kt - 20000

The profit function, P(t), of the sonic company is given by
P(t)= -0.02t^2 - 1.5kt - 20000
where t is the time in months after beginning operations, and k is a constant.
The profit begins to decrease after 75 months
Find the value of k.

Hey guys, i really need help with this question, so if possible can you show your working so i can follow how you got the answer please :)

So p'(75) = 0.
p'(t) = -.04t - 1.5k
-.04•75 - 1.5k = 0
1.5k = -3
k = -2

If you think about this function, it is a parabola that is opening downward (the -0.02 in front of t^2 tells me that). So, the peak is like the turning point, and that is where the profit most likely starts decreasing.

There is a formula for the x-value of the vertex: x = - b/(2a)

Here, b = -1.5k and a = -0.02. Anyway, you know the x-value at the vertex, that will be the 75. So fill in the equation.

75 = - (-1.5k)/(2*-0.02) and now solve for k. You should get k = -2.