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17 problems match “derivatives”. Clear

Smallest bound on derivative at zero ●●●○○ 50 ptsFind the smallest real number $A$ such that for every quadratic polynomial $f(x)$ satisfying $|f(x)| \leqslant 1$ for all $0 \leqslant x \leqslant 1$ , the inequality $f'(0) \leqslant A$ holds.
Minimum roots of g times g' ●●●○○ 50 ptsLet $f$ be a function defined for all $x \in \mathbb{R}$ that is discontinuous at exactly $3$ values of $x$ . Let $g$ be a differentiable function such that $f(x)g(x)$ is continuous for all $x \in \mathbb{R}$ . Find the minimum number of distinct real roots of the equation $g(x)g'(x)=0$ .
Cuboid of minimal surface area ●●●○○ 50 ptsA juice box is a rectangular box (cuboid) with volume $200\ \text{cm}^3$ ( $200$ ml). Find the length $l$ , width $w$ and height $h$ (in cm) of the box with the least possible surface area. Round each answer to five decimal places.
Minimal surface area for fixed volume ●●●○○ 50 ptsA regular right pentagonal prism has a volume of $278.3105$ cubic cm. Let $x$ be the side length of the regular pentagonal bases and $y$ the length (height) of the prism, in cm. Find the values of $x$ and $y$ that minimize the total surface area. Use the area of a regular pentagon $\approx 1.720477\,x^2$ . Round each value to four decimal places.
Stiffest rectangular beam from a log ●●●○○ 50 ptsThe stiffness of a beam is proportional to its width times the cube of its depth. Find the width and depth of the stiffest rectangular beam that can be cut from a cylindrical log of diameter 12 inches. Give the width and depth in inches. Round to three decimal places.
Function with midpoint averaging property ●●●○○ 50 ptsLet $f$ be a differentiable function satisfying $f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$ for all real $x$ and $y$ . Given that $f(0)=1$ and $f'(0)=-1$ , find $f(2)$ .
Maximum integer k in a derivative inequality ●●●●○ 100 ptsLet $f(x)=e^{x}-x-2$ . For $x>0$ , the inequality $(x-k)f'(x)+x+1>0$ always holds, where $k$ is an integer. Find the maximum value of $k$ .