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.
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(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.

Explanation:

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

Let $f'(x)$ be the derivative of $f(x)$.

Step2: Set $f'(x)=0$

Solve the equation $f'(x) = 0$ to find the critical - points. Let the critical - points be $x_1,x_2,\cdots,x_n$.

Step3: Test intervals

Choose test points in the intervals $(-\infty,x_1),(x_1,x_2),\cdots,(x_n,\infty)$. Evaluate $f'(x)$ at these test points.

• If $f'(x)>0$ on an interval, then $f(x)$ is increasing on that interval.
• If $f'(x)<0$ on an interval, then $f(x)$ is decreasing on that interval.
• A relative maximum occurs at a point $x = c$ where $f'(x)$ changes sign from positive to negative.
• A relative minimum occurs at a point $x = c$ where $f'(x)$ changes sign from negative to positive.

(a) The intervals where $f(x)$ is increasing are the intervals where $f'(x)>0$.
(b) The intervals where $f(x)$ is decreasing are the intervals where $f'(x)<0$.
(c) The $x$ - values of relative maxima are the $x$ - values where $f'(x)$ changes from positive to negative.
(d) The $x$ - values of relative minima are the $x$ - values where $f'(x)$ changes from negative to positive. Since we don't know the function $f(x)$, we can't give specific numerical answers. But the general process is as above.

Answer:

(a) Intervals where $f'(x)>0$
(b) Intervals where $f'(x)<0$
(c) $x$ - values where $f'(x)$ changes from positive to negative
(d) $x$ - values where $f'(x)$ changes from negative to positive