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# Week 10 Problem Set B.
## Reading.
**Please read and take notes on the following sections:**
Thomas Ch. 4.3 on Monotonic functions and the first derivative test.
Thomas Ch. 4.4 on Concavity and curve sketching.
as well as
Thomas Ch. 2.5 on Limits involving infinity: Asymptotes of a graph.
## Problems.
### Identifying extrema and intervals of increasing and decreasing.
In each of the following, (a) find the open intervals on which the function is increasing and decreasing, (b) identify the function's local maxima and local minima, and say where they occur, and (c) identify which of the extrema, if any, are absolute.
After you done it, check with a graphing calculator or DESMOS to verify your answers.
1. $f(x)=2x-x^{2}$, where $-\infty < x \le 2$.
2. $f(x) = (x+1)^{2}$, where $-\infty < x \le 0$
3. $f(x)=x^{2}-4x+4$, where $1\le x < \infty$
4. $f(x)=12x-x^{3}$, where $-3 \le x < \infty$
5. $\displaystyle f(x) = \frac{x^{3}}{3}-2x^{2}+4x$, where $0\le x < \infty$
6. $f(x)=\sqrt{25-x^{2}}$, where $-5\le x \le 5$
7. $\displaystyle f(x) = \frac{x-2}{x^{2}-1}$, where $0 \le x < 1$
### Theory and practice.
1. Sketch the graph of a differentiable function $y=f(x)$ through the point $(1,1)$ if $f'(1)=0$ and $f'(x) > 0$ for $x < 1$ and $f'(x) < 0$ for $x > 1$.
2. Sketch the graph of a differentiable function $y=f(x)$ through the point $(1,1)$ if $f'(1)=0$ and $f'(x) < 0$ for $x < 1$ and $f'(x) < 0$ for $x > 1$.
3. Sketch the graph of a differentiable function $y=f(x)$ through the point $(1,1)$ if $f'(1)=0$ and $f'(x) > 0$ for all $x \neq 1$.
4. Sketch the graph of a differentiable function $y=f(x)$ through the point $(1,1)$ if $f'(1)=0$ and $f'(x) < 0$ for all $x \neq 1$.
5. Determine the values of constants $a$ and $b$ so that $f(x)= ax^{2}+bx$ has an absolute maximum at the point $(1,2)$.
6. Determine the values of constants $a,b,c,d$ so that $f(x)=ax^{3}+bx^{2}+cx+d$ has a local maximum at the point $(0,0)$ and a local minimum at the point $(1,-1)$.
### Concavity.
For each of the following, (a) identify open intervals of increasing and decreasing, (b) identify open intervals of concave up and concave down, and (c) try to sketch it. (We will continue discussing sketching next week)
1. $y=6-2x-x^{2}$
2. $y=x(6-2x)^{2}$
3. $y=1-9x-6x^{2}-x^{3}$
4. $y=1-(x+1)^{3}$
5. $y=-x^{4}+6x^{2}-4$. Note: This can be re-expressed as $y=x^{2}(6-x^{2})-4$
There are some more ideas in this section that we will discuss next week, so please preview and take notes on the section Ch. 4.4 in the textbook for now.
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