Question

(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

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) \(-4\)