Concavity and Inflection Points

Which function is both always increasing and always concave down on its domain?

$f'(x)$ plot for a differentiable function is depicted as below. Answer following questions.

Find all the inflection points of $f$ shown in the plot.

$f'(x)$plot for a differentiable function is depicted as below. Answer following questions.

Find interval(s) at which $f$ is concave down.

Which one of the following statements is ALWAYS true for all differentiable functions $f(x)$ under the conditions given in the statement?

Consider the function $f(x) = x^4 - ax^3 + 2$ . For which values of $a$ does this function have no inflection points, or does such an $a$ not exist?

Consider the function $f(x) = x^4 - ax^3 + 2$ . For which values of $a$ does this function have no inflection points, or does such an $a$ not exist?

Given the following table of values, which statement(s) must always be true about $f\left(x\right)$? (Assume $f\left(x\right)$ is continuous)

Select all that apply.

Practice: Inflection Point

Find 𝑎 and 𝑏 values such tha$f\left(x\right)=ax^3+bx^2$ has an inflection point a$\left(-2,\ 48\right)$ .

Practice: Graph of a Derivative

Suppose that $f$ is a differentiable function on the interval $\left[-4,2\right]$. If the following graph represents the derivative of $f$, which of the following statements is correct?

How many inflection points does the function $f\left(x\right)=-\frac{1}{3x^2}-5x-e^{2x}$ have?

Find the point(s) of inflection and intervals of concave up and concave down of the graph of $y=\frac{x^4}{6}+\frac{3x^3}{2}-\frac{5x^2}{2}$.

Give the graph of $f'\left(x\right)$ below, which of the following statments must be true about the graph of $f\left(x\right)$?

(Select all that apply)

Find the inflection points of $f\left(x\right)=xe^x$

Which of the following statements is true about the function $f\left(x\right)=\left(x^2-3\right)e^{-x}$?

i.) The function has critical points at $x=\pm\sqrt{3}$

ii.) The function has a relative maximum at $x=3$

For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.

For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.

$f'(x)$ plot for a differentiable function is depicted below.

a) Find all critical point of $f$ in the range shown in the plot.

b) Determine which of the critical points found in the previous part are local max/min.

$f'(x)$ plot for a differentiable function is depicted below.

Find all the inflection points of $f$ shown in the plot.

If $$$f(x)$ is such that $f''(x)=(x^2-1)^2(x^2+1)$, then its inflection point(s) occur at

Determine the intervals of concavity of the function $f(x)=x^6-10x^4$ and the inflection points of its graph.

For the following function, determine the intervals in which the function is increasing or decreasing, its critical points, and the intervals in which the function is concave upwards or downwards.

Determine where the following function is concave up/down: $f(x)=xe^x$

$\textbf{Q.}$ Determine the interval(s) on which the graph of $\displaystyle f(x)=\frac{1}{x^{2}+4}$ is concave up or down, and the inflection points.

The graph of the derivative $f^{'}(x)$ is shown

Where are the local minimums of $f(x)$located? Also list the intervals for which f is concave up or down.

The graph of the derivative $f^{'}(x)$ is shown

Where are the local minimums of $f(x)$located? Also list the intervals for which f is concave up or down.

The graph of the derivative $f^{'}(x)$ is shown

List the intervals for which f is concave up or down.

Determine where the following function is concave up/down: $f(x)=xe^x$

Find all inflection points for the function $f(x)=2x^3-2x^2+1$

Given $f(x)=2-2x+\frac{1}{3}x^3$, find

• the intervals of increase and decrease
• the local maximum and minimum values

Find the intervals of increase/decrease and the intervals of concavity for $f(x)=\ln(x^2-1)$

Practice Question: Max, Min, and Inflection Points

Given the following table of values, which statement(s) must always be true about $f\left(x\right)$?

i.) $f\left(x\right)$ has 3 critical points

Practice Question: Inflection Point

Find 𝑎 and 𝑏 values such tha$f\left(x\right)=ax^3+bx^2$ has an inflection point a$\left(-2,\ 48\right)$ .

For a function $f$, its 2nd derivative is $f''\left(x\right)=8x^6-40x^5$

Find the values of x for which $f$ has a point of inflection

$\displaystyle f(x)=\frac{1}{x^2+1}$.

The first and the second derivatives are

$\displaystyle f(x)=\frac{1}{x^2+1}$.

The first and the second derivatives are

Which function is both always increasing and always concave down on its domain? (Select all that apply)

Given this table of values, which of the following statements must NOT be true?

$\begin{array}{|c|c|c|c|} \hline &f(x)&f'(x)&f''(x)\\ \hline x=0&3&1&3\\ x=1&0&0&-1\\ x=2&0&-2&0\\ \hline \end{array}$

The following is the graph of $f''\left(x\right)$, which of the following statements (if any) is correct?

Let $f''(x)=\frac{6x^2\left(x-1\right)}{(x^2-6)^{3/2}(x-3)^3}$

Practice: Graph of a Derivative

Suppose that $f$ is a differentiable function on the interval $(-4,2)$. If the following graph represents the derivative of $f$, which of the following statements is correct?

$\textbf{Q.}$ Determine the interval(s) on which the graph of $\displaystyle f(x)=\frac{1}{x^{2}+4}$ is concave up or down, and the inflection points.

Determine where the following function is concave up/down: $f(x)=xe^x$

Practice: Graph of a Derivative

Suppose that $f$ is a differentiable function on the interval $\left[-4,2\right]$. If the following graph represents the derivative of $f$, which of the following statements is correct?

Practice: Inflection Point

Find 𝑎 and 𝑏 values such tha$f\left(x\right)=ax^3+bx^2$ has an inflection point a$\left(-2,\ 48\right)$ .

Given the following table of values, which statement(s) must always be true about $f\left(x\right)$? (Assume $f\left(x\right)$ is continuous)

Select all that apply.

How many inflection points does the function $f\left(x\right)=-\frac{1}{3x^2}-5x-e^{2x}$ have?

$f'(x)$plot for a differentiable function is depicted as below. Answer following questions.

Find interval(s) at which $f$ is concave down.

Given this table of values, which of the following statements must NOT be true?

$\begin{array}{|c|c|c|c|} \hline &f(x)&f'(x)&f''(x)\\ \hline x=0&3&1&3\\ x=1&0&0&-1\\ x=2&0&-2&0\\ \hline \end{array}$

Determine the intervals of concavity of the function $f(x)=x^6-10x^4$ and the inflection points of its graph.