Suppose that f(x) = ln(5 + x^2) Use interval notation to indicate where f(x) is concave up, concave down.

Suppose that f(x) = ln(5 + x^2) Use interval notation to indicate where f(x) is concave up, concave down.

Question
Analyzing functions
asked 2020-11-08
Suppose that \(\displaystyle{f{{\left({x}\right)}}}={\ln{{\left({5}+{x}^{{2}}\right)}}}\)
Use interval notation to indicate where f(x) is concave up, concave down.

Answers (1)

2020-11-09
\(\displaystyle{f{{\left({x}\right)}}}={\ln{{\left({5}+{x}^{{2}}\right)}}}\)
\(\displaystyle{f}{1}{\left({x}\right)}=\frac{{{2}{x}}}{{{5}+{x}^{{2}}}}\)
\(\displaystyle{f}{2}{\left({x}\right)}=\frac{{{2}{\left({5}–{x}^{{2}}\right)}}}{{\left({5}+{2}^{{2}}\right)}^{{2}}}={0}\)
\(\displaystyle{x}=\sqrt{{5}},-\sqrt{{5}}\)
Downward \(\displaystyle{\left(-\infty,-\sqrt{{5}}\right)}\cup{\left(\sqrt{{5}},\infty\right)}\)
Upwards \(\displaystyle{\left(-\sqrt{,}\sqrt{{5}}\right)}\)
0

Relevant Questions

asked 2021-01-02
Sketch the graph of the function \(\displaystyle{f{{\left({x}\right)}}}=-{x}^{{3}}+{3}{x}^{{2}}–{7}\). List the coordinated of where extrema or points of inflection occur. State where the function is increasing or decreasing as well as where it is concave up or concave down.
asked 2021-05-06
A spring is mounted vertically on the floor. The mass of the spring is negligible. A certain object is placed on the spring to compress it. When the object is pushed further down by just a bit and then released, one up/down oscillation cycle occurs in 0.250 s. However, when the object is pushed down by 5.00 X 10^-2 m to point P and then released, the object flies entirely off the spring. To what height above point P does the object rise in the absence of air resistance?
asked 2021-05-03
Suppose \(\int_{-2}^{2}f(x)dx=4,\int_{2}^{5}f(x)dx=3, \int_{-2}^{5}g(x)dx=2\).
Is the following statement true?
\(f(x)\leq g(x)\) on the interval \(-2\leq x\leq 5\)
asked 2021-04-07
Ok, If a bobsled makes a run down an ice track starting at 150m vertical distance up the hill and there is no friction, what isthe velocity at the bottom of the hill?
I know that the initial velocity here is 0 because it isstarting from rest. And this problem deal with theconservation of energy. But, I don't know where to go fromhere.
asked 2021-05-12
Find the absolute maximum and absolute minimum values of f on the given interval and state where those values occur: \(f(x)=x^{3}-3x^{2}-9x+25, [-5,10]\)
asked 2021-03-25
A skier wears a jacket filled with goose down that is 15mmthick. Another skier wears a wool sweater that is 5.0mmthick. Both have the same surface area. Assuming thetemperature difference between the inner and outer surfaces of eachgarment is the same, calculate the ratio (wool./goose down) of theheat lost due to conduction during the same time interval.
asked 2021-05-05
The bulk density of soil is defined as the mass of dry solidsper unit bulk volume. A high bulk density implies a compact soilwith few pores. Bulk density is an important factor in influencing root development, seedling emergence, and aeration. Let X denotethe bulk density of Pima clay loam. Studies show that X is normally distributed with \(\displaystyle\mu={1.5}\) and \(\displaystyle\sigma={0.2}\frac{{g}}{{c}}{m}^{{3}}\).
(a) What is thedensity for X? Sketch a graph of the density function. Indicate onthis graph the probability that X lies between 1.1 and 1.9. Findthis probability.
(b) Find the probability that arandomly selected sample of Pima clay loam will have bulk densityless than \(\displaystyle{0.9}\frac{{g}}{{c}}{m}^{{3}}\).
(c) Would you be surprised if a randomly selected sample of this type of soil has a bulkdensity in excess of \(\displaystyle{2.0}\frac{{g}}{{c}}{m}^{{3}}\)? Explain, based on theprobability of this occurring.
(d) What point has the property that only 10% of the soil samples have bulk density this high orhigher?
(e) What is the moment generating function for X?
asked 2021-05-21
Find the absolute maximum and absolute minimum values of f over the interval. \(f(x)=(\frac{4}{x})+\ln(x^{2}), 1\leq x\leq 4\)
asked 2020-11-11

Use exponential regression to find a function that models the data. \(\begin{array}{|c|c|} \hline x & 1 & 2 & 3 & 4 & 5 \\ \hline f(x) & 14 & 7.1 & 3.4 & 1.8 & 0.8 \\ \hline \end{array}\)

asked 2021-02-04
Find all values x = a where the function is discontinuous.
\(\displaystyle{f{{\left({x}\right)}}}={8}{x}^{{2}}+{8}{x}+{4}.\)
...