Let [tex]\( f(x) = x^3 + 3x^2 - 9x + 21 \)[/tex].
(a) To find the first derivative [tex]\( f'(x) \)[/tex], we use the rules of differentiation.
[tex]\[ f'(x) = \frac{d}{dx}(x^3 + 3x^2 - 9x + 21) = 3x^2 + 6x - 9 \][/tex]
Thus, [tex]\( f'(x) = 3x^2 + 6x - 9 \)[/tex].
(b) To find the second derivative [tex]\( f''(x) \)[/tex], we differentiate [tex]\( f'(x) \)[/tex].
[tex]\[ f''(x) = \frac{d}{dx}(3x^2 + 6x - 9) = 6x + 6 \][/tex]
Thus, [tex]\( f''(x) = 6x + 6 \)[/tex].
(c) To determine the interval where [tex]\( f \)[/tex] is increasing, we solve [tex]\( f'(x) > 0 \)[/tex]. First, find the critical points by setting [tex]\( f'(x) = 0 \)[/tex].
[tex]\[ 3x^2 + 6x - 9 = 0 \][/tex]
The solutions are:
[tex]\[ x = -3 \][/tex]
[tex]\[ x = 1 \][/tex]
An interval of increasing is identified by checking the sign of [tex]\( f'(x) \)[/tex] between critical points. There are no intervals where [tex]\( f'(x) > 0 \)[/tex]. Thus,
[tex]\[ \text{interval of increasing} = [] \][/tex]
(d) To determine the interval where [tex]\( f \)[/tex] is decreasing, we solve [tex]\( f'(x) < 0 \)[/tex]. We already found our critical points: [tex]\( x = -3 \)[/tex] and [tex]\( x = 1 \)[/tex]. We check the sign of [tex]\( f'(x) \)[/tex] in the intervals determined by these points:
[tex]\[ \text{interval of decreasing} = [(-3, 1)] \][/tex]
(e) To determine the interval where [tex]\( f \)[/tex] is concave downward, we solve [tex]\( f''(x) < 0 \)[/tex]. Critical points come from solving [tex]\( f''(x) = 0 \)[/tex], and:
[tex]\[ 6x + 6 = 0 \][/tex]
The solution is:
[tex]\[ x = -1 \][/tex]
Checking intervals on either side of [tex]\( x = -1 \)[/tex], there are no intervals where [tex]\( f''(x) < 0 \)[/tex]. Thus,
[tex]\[ \text{interval of downward concavity} = [] \][/tex]
(f) To determine the interval where [tex]\( f \)[/tex] is concave upward, we solve [tex]\( f''(x) > 0 \)[/tex]. We already found the critical point [tex]\( x = -1 \)[/tex], and there are no intervals where [tex]\( f''(x) > 0 \)[/tex]. Thus,
[tex]\[ \text{interval of upward concavity} = [] \][/tex]
In summary:
(a) [tex]\( f'(x) = 3x^2 + 6x - 9 \)[/tex]
(b) [tex]\( f''(x) = 6x + 6 \)[/tex]
(c) interval of increasing [tex]\( = [] \)[/tex]
(d) interval of decreasing [tex]\( = [(-3, 1)] \)[/tex]
(e) interval of downward concavity [tex]\( = [] \)[/tex]
(f) interval of upward concavity [tex]\( = [] \)[/tex]
