5. Graphical Applications of Derivatives

For the function $y=3x^4-2x$ defined on the closed interval $[0,1]$, at what value of $x$ does the minimum value of $y$ occur?

Find the minimum and maximum values of the function $f(x,y)=x^2+y^2$ subject to the constraint $x+y=4$.

Two numbers sum to $100$. What is the smallest possible value of the sum of one number and the square of the other number?

Given the region bounded by the x-axis, the y-axis, and the line $y=4-x$ in the first quadrant, determine the $x$ and $y$ coordinates of the centroid of the shaded area.

Suppose the product of two positive real numbers is $16$. Which pair of numbers has the smallest possible sum?

Your café sells lattes for \$4 each to 100 customers per day. For every \$1 increase in price, you would lose 20 customers. Find the price that maximizes revenue. Hint: The # of items sold is based on the number of customers.

A poster is set to have a total area of 1150 cm², with 2-cm margins on the sides and the top, and a 3-cm margin at the bottom. What dimensions will maximize the printed area?