Intro to Extrema: Videos & Practice Problems

Consider a cuboidal 8 m³ container with a square base and no top: each side of the base measures a, and the height of the container is b. Sketch the graph of the function that depicts the surface area of the container S(a) for a > 0.

Using the graph, estimate the value of a that minimizes the surface area, and round your answer to 2 decimal places.

Determine the intervals on which the function $f(x) = \(\sin\)^2(x)$ is increasing or decreasing on the interval $\(\left\)[-\(\frac{\pi}{2}\), \(\frac{\pi}{2}\]\right\)]$.

Determine the critical points of the function $f\left(x\right)=\frac{x}{x^2+4}$.

Determine the critical points of the function $f\left(x\right)=2\ln \left(x\right)-x^2$.

Determine the intervals of increasing/decreasing and the critical points of the function using the graph of $f^{\prime }\left(x\right)$.

A projectile is launched vertically upwards and its height above the ground after $t$ seconds is given by the function $h(t) = 50t - 5t^2$, where $0\(\le\) t<10$. At what time is the projectile at its maximum height?

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?

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 $h(t)=\(\begin{cases}\]\frac\)95t^2,\(\text{ if }\)0\(\le\) t<60\\ -\(\frac\)95\(\left\)(t^2-240t+120\(\right\)),\(\text{if }\)60\(\le\) t<120\(\end{cases}\)$ , 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^{\(\prime\)}(x)=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.

Suppose the domain of $f\left(x\right)$ is $(-1,1)$. Determine the intervals where $f\left(x\right)$ is increasing and decreasing using the graph of its derivative $f^{\prime }\left(x\right)$.

Examine the function $f\left(x\right)$ plotted on the interval $[0,4]$. Identify the $x$-values where the function attains its absolute maximum and minimum.

Examine the function $f\left(x\right)$ plotted on the interval $[0,4)$. Identify the $x$-values where the function attains its absolute maximum or minimum.

Find the intervals where the function $g\left(x\right)=(2x-3{)}^2$ is increasing and decreasing.

Examine the function $f\left(x\right)$ plotted on the interval $[1.5,3]$. Identify the $x$-values where the function attains its absolute maximum or minimum.

Find the intervals where the function $g\left(x\right)=\frac{3e^x}{e^{2x}+1}$ is increasing and decreasing.

Examine the function $f\left(x\right)$ plotted on the interval $[1,5]$. Identify the $x$-values where the function attains its absolute maximum or minimum.

Find the intervals where the function $g\left(x\right)=x^{\frac{3}{2}}(x-3)$ is increasing and decreasing.

Determine the critical points of the function $g\left(x\right)=2x^2+5x-3$.

Determine the critical points of the function $h\left(x\right)=\frac{x^3}{3}-12x$.

Determine the critical points of the function $h\left(x\right)=4x^3+2x^2-8x$.

Determine the $x$-coordinates of the critical points of the function $f\left(x\right)=x^3-6bx$, where $b$ is a constant and $b\ne 0$.

Determine the critical points of the function $f\left(x\right)=4e^x+4e^{-x}$.

Determine the critical points of the function $f\left(x\right)=2x^2\sqrt{2x-4}$.

Determine the $x$-coordinate of the critical point of the function $f\left(x\right)=2x\sqrt{x-2b}$, where $b$ is a constant and $b>0$.

Determine the intervals on which the function $f\left(x\right)=\sqrt{25-x^2}+{\sin }^{-1}\left(\frac{x}{5}\right)$ is increasing and decreasing.

Determine the $x$-coordinate of the critical point of the function $f\left(x\right)=\frac{1}{4}{x}^4-b^{3}x$, where $b$ is a constant and $b\ne 0$.

Determine the intervals on which the function $f\left(x\right)=5x\ln \left(5x\right)-5x+5$ is increasing and decreasing on the interval $(0,\infty )$.

Determine the intervals on which the function $g\left(x\right)=3x^4-16x^3+24x^2$ is increasing and decreasing.

Determine the intervals on which the function $g\left(x\right)=x^3-3\ln \left(x\right)$ is increasing and decreasing.

Determine the intervals on which the function $g\left(x\right)=-8x^4+x^2+17$ is increasing and decreasing.