For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing.

35. [tex]f(x) = x^4 - 4x^3 + 5[/tex]

36. [tex]h(x) = x^5 + 5x^4 + 10x^3 + 10x^2 - 1[/tex]

37. [tex]g(t) = t\sqrt{t+3}[/tex]

38. [tex]k(t) = 3t^{\frac{2}{3}} - t[/tex]

39. [tex]m(x) = x^4 + 2x^3 - 12x^2 - 10x + 4[/tex]

40. [tex]n(x) = x^4 - 8x^3 + 18x^2 - 6x + 2[/tex]

Sure! Let's go through the solution step-by-step for the given question:

Eddie is arranging cards numbered from 1 to 10 in a row. To determine the number of possible arrangements, we need to find all the permutations of these 10 cards.

The total number of permutations of [tex]$ n $[/tex] distinct items is given by the factorial of [tex]$ n $[/tex], denoted as [tex]$ n! $[/tex]. In this scenario, [tex]$ n $[/tex] is 10.

### Step-by-step calculation:

1. Understanding Factorial:
- The factorial of a number [tex]$ n $[/tex], denoted as [tex]$ n! $[/tex], is the product of all positive integers less than or equal to [tex]$ n $[/tex].
- For example, [tex]$ 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 $[/tex].

2. Calculate 10!:
- To find the number of ways to arrange 10 cards, we need to calculate [tex]$ 10! $[/tex].
- [tex]$ 10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 $[/tex].

3. Perform the Multiplication:
- Calculate the product step by step:
- [tex]$ 10 \times 9 = 90 $[/tex]
- [tex]$ 90 \times 8 = 720 $[/tex]
- [tex]$ 720 \times 7 = 5040 $[/tex]
- [tex]$ 5040 \times 6 = 30240 $[/tex]
- [tex]$ 30240 \times 5 = 151200 $[/tex]
- [tex]$ 151200 \times 4 = 604800 $[/tex]
- [tex]$ 604800 \times 3 = 1814400 $[/tex]
- [tex]$ 1814400 \times 2 = 3628800 $[/tex]
- [tex]$ 3628800 \times 1 = 3628800 $[/tex]

Thus, the total number of possible arrangements of the 10 cards is [tex]$ 3,628,800 $[/tex].