Re: mattie_j37890's quetion at Yahoo! Answers regarding approximating critical values
Hello mattie_j37890,
We are given the function:
$$S(t)=-0.00003237t^5+0.0009037t^4-0.008956t^3+0.03629t^2-0.04467t+0.4438$$
First, let's look at a plot of this function just so we have some idea of what it looks like over the relevant domain:
View attachment 944
We can see that there are 4 relative extrema in the interval $0\le t\le10$
To find where they are, that is for what values of $t$ do they exist, we want to equate the first derivative to zero:
$$S'(t)=-0.00016185t^4+0.0036148t^3-0.026868t^2+0.07258t-0.04467=0$$
Let's look at a plot of the derivative to get some idea of where the roots are:
View attachment 945
We may use Newton's method to approximate these roots. We may write:
$$f(t)=-0.00016185t^4+0.0036148t^3-0.026868t^2+0.07258t-0.04467=0$$
$$f'(t)=-0.0006474t^3+0.0108444t^2-0.053736t+0.07258$$
And so we have the recursion:
$$t_{n+1}=t_{n}-\frac{f\left(t_{n} \right)}{f'\left(t_{n} \right)}$$
We see the first root is near $t=1$, and so we may compute:
$$t_0=1$$
$$t_1\approx0.845220550257$$
$$t_2\approx0.857337077417$$
$$t_3\approx0.857416179268$$
$$t_4\approx0.857416182626$$
$$t_5\approx0.857416182626$$
We see the second root is near $t=5$, and so we may compute:
$$t_0=5$$
$$t_1\approx4.53064243449$$
$$t_2\approx4.60497652706$$
$$t_3\approx4.60659458701$$
$$t_4\approx4.606595386$$
$$t_5\approx4.606595386$$
We see the third root is near $t=7$, and so we may compute:
$$t_0=7$$
$$t_1\approx7.3250339402$$
$$t_2\approx7.30609727792$$
$$t_3\approx7.30607003007$$
$$t_4\approx7.30607003001$$
$$t_5\approx7.30607003001$$
We see the fourth root is near $t=10$, and so we may compute:
$$t_0=10$$
$$t_1\approx9.66222062004$$
$$t_2\approx9.57077528295$$
$$t_3\approx9.56421133076$$
$$t_4\approx9.56417851958$$
$$t_5\approx9.56417851876$$
$$t_6\approx9.56417851876$$
And so, we have the four roots of the first derivative approximated by:
$$t\approx0.857416182626,\,4.606595386,\,7.30607003001,\,9.56417851876$$
Now, to find the absolute extrema on the given closed interval, we need to evaluate the function at the endpoints and at the critical values:
$$S(0)=0.4438$$
$$S(0.857416182626)=0.4270063563925$$
$$S(4.606595386)=0.47243084424924303$$
$$S(7.30607003001)=0.462862495347419$$
$$S(9.56417851876)=0.47195253062506465$$
$$S(10)=0.4701$$
And so rounding to 3 decimal places, we find:
The absolute minimum occurs when $$t\approx0.857$$
The absolute maximum occurs when $$t\approx4.607$$
