(a) find the interval(s) where f(x) is increasing. (b) find the interval(s) where f(x) is decreasing. (c)…

(a) find the interval(s) where f(x) is increasing. (b) find the interval(s) where f(x) is decreasing. (c) find the x - value(s) of all relative maxima of f(x). if there are none, enter none. if there are multiple relative maxima, separate the values with commas. (d) find the x - value(s) of all relative minima of f(x). if there are none, enter none. if there are multiple relative minima, separate the values with commas. f(x)=x^{2}+8x - 9

(a) find the interval(s) where f(x) is increasing. (b) find the interval(s) where f(x) is decreasing. (c) find the x - value(s) of all relative maxima of f(x). if there are none, enter none. if there are multiple relative maxima, separate the values with commas. (d) find the x - value(s) of all relative minima of f(x). if there are none, enter none. if there are multiple relative minima, separate the values with commas. f(x)=x^{2}+8x - 9

Answer

Explanation:

Step1: Find the derivative of (f(x))

Given (f(x)=x^{2}+8x - 9), using the power - rule ((x^n)^\prime=nx^{n - 1}), we have (f^\prime(x)=2x + 8).

Step2: Find the critical points

Set (f^\prime(x)=0), so (2x+8 = 0). Solving for (x), we get (2x=-8), then (x=-4).

Step3: Determine where (f(x)) is increasing or decreasing

We consider the intervals separated by the critical point (x = - 4). - For the interval ((-\infty,-4)), let's take a test - point, say (x=-5). Then (f^\prime(-5)=2\times(-5)+8=-2<0), so (f(x)) is decreasing on the interval ((-\infty,-4)). - For the interval ((-4,\infty)), let's take a test - point, say (x = 0). Then (f^\prime(0)=2\times0 + 8=8>0), so (f(x)) is increasing on the interval ((-4,\infty)).

Step4: Find relative extrema

Since (f(x)) changes from decreasing to increasing at (x=-4), (f(x)) has a relative minimum at (x=-4). There are no relative maxima.

Answer:

(a) ((-4,\infty)) (b) ((-\infty,-4)) (c) NONE (d) (-4)