Applications of Differentiation
Extension Problems
Consider $f\left(x\right)=x^3+ax^2+bx+c$.
It is known that $f\left(x\right)$ is increasing for all $x\in \mathbb{R}$ and $f\left(1\right)=0$.
Find the maximum value of $c$.
*Give your answer as a simplified fraction
It is known that $f\left(x\right)$ is increasing for all $x\in \mathbb{R}$ and $f\left(1\right)=0$.
Find the maximum value of $c$.
*Give your answer as a simplified fraction
Find the maximum and minimum values of the function $f\left(x\right)=x+\sqrt{1-4x^2}$.
*Give your answers as exact values
*Give your answers as exact values
$max=$
$min=$
Consider the graphs of $f\left(x\right)=x^2$ and $g\left(x\right)=x^3$.
There are two lines that are tangent to both graphs. One is $y=0$. Find the equation of the other line.
*Give your answer in the form $ax+by+d=0$ where $a, b$ and $d$ are integers.
There are two lines that are tangent to both graphs. One is $y=0$. Find the equation of the other line.
*Give your answer in the form $ax+by+d=0$ where $a, b$ and $d$ are integers.
Find the coordinates of the global minimum on the graph of $\displaystyle y=\frac{2^x \left(x-1\right)^{\left(x-1\right)}}{x^x}$ for $x\gt1$.
*Hint: take the natural log of both sides and use implicit differentiation
*Hint: take the natural log of both sides and use implicit differentiation
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Find the maximum area of a rectangle that can be inscribed between the graph of $f\left(x\right)=x\left(x-1\right)^2$ and the $x$-axis from $0 \lt x \lt 1$, as shown in the diagram.
*Give your answer as a simplified fraction in the form $\frac{a\sqrt{b}-c\sqrt{d}}{e}$ where $a,b,c,d,e\in \mathbb{Z}^+$
A person is standing $6$ meters away from a large movie screen. The height of the movie screen is $6$ meters and the bottom of the movie screen is $2$ meters above the height of the person's eyes. If the person walks towards the screen at a speed of $0.3$ meters per second, find the rate at which the angle formed by the person's eyes and the top and bottom of the screen changes in radians per second.
radians per second