Answer: t=10, or in 2005
Explanation: the point of inflection refers to when a function switches concavity (i.e. up or down). If the second derivative of f is positive, the graph is concave up, and if it is negative, the graph is concave down. This means that the graph switches concavity when the second derivative equals zero.
To find the second derivative (f’’(t)), we shall first find f’(t), or the first derivative. Because each term is separated by addition/subtraction, we can take the derivative of each term separately. Using the power rule, ((d/dx)(-0.1t^3)) = -0.1(3)(t)^(3-1)= -0.3t^2. For d/dx of 3t^2, use the power rule again: 3(2)t^(2-1) = 6t. The derivative of 500 is 0 because it is a constant. f’(t) = -0.3t^2 + 6t.
Now we shall do this process again to find the second derivative. For the first term, d/dx of -0.3t^2 = -0.3(2)t^(2-1) = -0.6t. For 6t, d/dx equals 6(1)t^(1-1) = 6. f”(t) = -0.6t + 6
Now that we have the second derivative, we set it equal to zero and solve for t. 0 = -0.6t + 6, -6 = -0.6t, t = 10. Because t = 10 years, the point of inflection occurs in 1995 + 10 = 2005.