Question

graph the equation below using a calculator and point - by - point plotting. indicate the increasing and decreasing intervals.
y = - 3 ln x
choose the correct graph below.
where is the graph increasing or decreasing? select the correct choice below and fill in any answer box(es) in your choice, if necessary.
a. the graph increases on and decreases on
(type your answer in interval notation.)
b. the graph decreases on but does not increase.
(type your answer in interval notation.)
c. the graph increases on but does not decrease.
(type your answer in interval notation.)
d. the graph neither increases nor decreases

Explanation:

Step1: Analyze domain

The domain of $y = - 3\ln x$ is $x>0$ since the natural - logarithm function $\ln x$ is only defined for positive real numbers.

Step2: Take the derivative

The derivative of $y=-3\ln x$ using the formula $\frac{d}{dx}(\ln x)=\frac{1}{x}$ is $y'=-\frac{3}{x}$.

Step3: Determine increasing and decreasing intervals

For $x > 0$, $y'=-\frac{3}{x}<0$. When the derivative of a function is negative on an interval, the function is decreasing on that interval. So the function $y = - 3\ln x$ is decreasing on the interval $(0,\infty)$.

Answer:

B. The graph decreases on $(0,\infty)$ but does not increase.