The derivative of power function given by

\(\frac{d}{dx} (x^{n}) = nx^{n - 1}\)

Procedure to find the extrema of the continuous function f on closed interval [a, b].

Step 1: Find the derivative of the function f.

Step 2: Find the critical points of f in the open interval (a, b).

Step 3: Determine the value of f at each of the critical numbers in the open interval (a,b).

Step 4: Determine the value of f at each of the end-points a and b.

Step 5: The least of these values is the minimum and the greatest is the maximum.

Calculation:

Consider the function \(P = P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8,\)

Step 1. Determine first derivative of the function \(P = P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8, 0\)

\(P` = -0.0069t^{2} + 0.03t + 2.83\)

Step 2. Find out the critical points of P. It occurs when \(P` = 0\) or P` underfined.

\(P` = 0\)

\(-0.0069t^{2} + 0.03t + 2.83 = 0\)

Apply quadratix formula as,

\(t=\frac{-0.03 \pm \sqrt{0.03}^{2}-4(-0.0069)(2.83)}{2(-0.0069)}\)

\(=\frac{-0.03 \pm \sqrt{0.0009 + 0.078108}}{-0.0138}\)

\(=\frac{-0.03 \pm \sqrt{0.079008}}{-0.0138}\)

\(=\frac{-0.03 \pm 0.281}{-0.0138}\)

Further solving,

\(t =\frac{-0.03 \pm 0.281}{-0.0138}\)

\(= -18.19\)

And,

\(t =\frac{-0.03 - 0.281}{-0.0138}\)

\(= 22.53\)

This gives,

\(t = -18.19, 22.53\)

For end-point \(t = 0,\)

Substitute \(t = 0\) in the function \(P = -0.00232t^{3} + 0.0151y^{2} + 2.83t + 281.8,\)

\(P = -0.0023(0)^{3} + 0.015(0)^{2} + 2.83(0) + 281.8,\)

\(= 281.8\)

For end-point \(t = 13,\)

Substitute \(t = 13\) in the function \(P = -0.0023^{3} + 0.015^{2} + 2.83t + 281.8,\)

\(P = -0.0023(13)^{3} + 0.015(13)^{2} + 2.83(13) + 281.8,\)

\(= -0.0023 (2197) + 0.015 (169) + 3.83(13) + 281.8\)

\(= -5.531 + 2.535 + 36.79 + 281.8\)

\(= 316.1\)

Use all the information to form the table as,

\(\begin{array}{|c|c|} \hline t-value & s=0 & r=13 \\ \hline P & 281.8 & 316.1 \\ \hline Conclusion & Minimum & Maximum \\\hline \end{array}\)

From the table, it can be concluded that the minimum population was 281.8 million in 2000 and the maximum population was 316.1 million in 2013.