Intro to Extrema: Videos & Practice Problems

Determine the intervals on which the function $f\left(x\right)=\frac{x^4}{4}-5x^3+28x^2+68$ is increasing and decreasing.

Determine the $x$-values which represent a local minimum and local maximum value of $f\left(x\right)$ using the given graph of $f^{\prime }\left(x\right)$?

Determine the intervals on which the function $f\left(x\right)=x^2{e}^{-\frac{x^2}{4}}$ is increasing and decreasing.

Determine the intervals on which the function $f\left(x\right)={\cos }^{-1}\left(\frac{2x}{x^2+3}\right)$ is increasing and decreasing.

A projectile is launched vertically upwards and its height above the ground after $t$ seconds is given by the function $h\left(t\right)=50t-5t^2$, where . At what time is the projectile at its maximum height?

At what time does the ball reach its maximum height if it is thrown upward from the top of a $50$ $\text{m}$ high building with an initial speed of $20$ $\text{m/s}$, and its height above the ground is given by the equation $s\left(t\right)=-10t^2+20t+50$?

Evaluate the behavior of the function $p\left(x\right)=x^3$ on the interval $(-\infty ,\infty )$ to determine if it has a local minimum.

Which of the following properties correctly describes the given graph?

On the interval $[-2,2]$, determine the absolute maximum and minimum values of the function $p\left(x\right)=4x^3-12x^2+8x$.

Find the local extrema and inflection points of the function $f\left(x\right)$ if $f\left(x\right)=\frac{1}{6x^4+10},f^{\prime }\left(x\right)=-\frac{6x^3}{{(3x^4+5)}^2}$, and $f^{\prime \prime }\left(x\right)=\frac{90x^2(x^4-1)}{{(3x^4+5)}^3}$.

A container is being filled with a liquid at a rate that changes over time. The volume of the liquid in the container after $t$ seconds is given by $V=200(50-t{)}^2$ cubic meters. At what time is the magnitude of the flow rate into the container a maximum?

Examine the function $f\left(x\right)$ plotted on the interval $[0.5,5.4]$. Identify the $x$-values where the function attains its local and absolute extrema.

Examine the function $g\left(x\right)$ plotted on the interval $[-3.4,3.2]$. Identify the $x$-values where the function attains its local and absolute extrema.

Let $m$ be a differentiable function on the real numbers with a local maximum at $x=5$, where $m\left(5\right)=-1$. Define $n\left(x\right)=3xm\left(x\right)+5$ and $o\left(x\right)=2xm\left(x\right)+3x+5$. Calculate $n\left(5\right)$, $o\left(5\right)$, $n^{\prime }\left(5\right)$, and $o^{\prime }\left(5\right)$.

Let $m$ be a differentiable function on the real numbers with a local maximum at $x=5$, where $m\left(5\right)=-1$. Define $n\left(x\right)=3xm\left(x\right)+5$ and $o\left(x\right)=2xm\left(x\right)+3x+5$. Find out if either $n$ and $o$ have a local extreme value at $x=5$.

Given the function $g\left(x\right)=2x^4-8x^2+4$, identify the $x$-value at which $g\left(x\right)$ has a local extremum.

Determine the properties that describe the graph of $f\left(x\right)$ on $[1,6.5]$.

Do the following properties correctly describe the graph of $f\left(x\right)$ on $[0,8]$? Justify if incorrect.

$f^{\prime }\left(2\right)$ and $f^{\prime }\left(4\right)$ are undefined, $f^{\prime }\left(6\right)=0$, $f\left(x\right)$ has local minima at $x=4$ and $x=6$, $f\left(x\right)$ has a local maximum at $x=2$, $f\left(x\right)$ has an absolute maximum at $x=0$, $f\left(x\right)$ has an absolute minimum at $x=6$

A farmer is monitoring the growth of a crop over a $120$-day period. The height of the crop in centimeters is given by the function , where $t$ is the time in days since initially planting their crops. When is the growth rate of the crop at a maximum?

Given the derivative $f^{\prime }\left(x\right)=8\cos x-4\sin 2x$ on the interval $[0,2\pi ]$, identify the $x$-coordinates of the local maxima and minima of $f$, as well as the intervals where $f$ is increasing or decreasing.

Determine the extreme values (absolute and local) of the function $g\left(x\right)=4x^2-16x+15$ over its natural domain, and where they occur.

Determine the extreme values (absolute and local) of the function $g\left(x\right)=\sqrt{4-x^2}$ over its natural domain, and where they occur.

Identify the extreme values (absolute and local) of the function $h\left(x\right)=\sqrt{x^2+4x+3}$ over its natural domain.

Determine the extreme values (absolute and local) of the function $g\left(x\right)=\frac{3x+2}{x^2+4x+5}$ over its natural domain.

Based on the graph of $j^{\prime }\left(x\right)$, determine where the function $j\left(x\right)$ has local extrema and classify them as either a relative maximum or minimum. Also, determine the intervals where $j\left(x\right)$ is increasing/decreasing.

A student is studying the behavior of a function $g\left(t\right)$ defined by $g\left(t\right)=-5t^2$. For which values of $t$ is $g^{\prime }\left(t\right)$ positive, zero, or negative?

For the function $h\left(x\right)=\frac{5}{x}$, identify the intervals of $x$-values where the function is increasing and decreasing.

What value of $c$ makes $h\left(z\right)=3z^2+\frac{c}{z}$ have a point of inflection at $z=-2$?

Determine the values of $p$ and $q$ that make the function $j\left(x\right)=x^3+p{x}^2+qx+1$ have a local maximum at $x=-3$ and a point of inflection at $x=2$.

Identify the local extreme values of the function $p\left(x\right)=(x+4{)}^2-5$ in the domain .