1. Describe the graph and critical values and characteristics of
f(x) = -(2)^x +1
2. What is the solution set? What steps did you take to determine the solution set?
x^2+9x+20/x^2-x-20 ≥ 0
3. How do you use the Unit Circle to find the cos 210° in exact form? No decimals.
4. After the parent cosine functions are:
Vertically compressed by one half
Horizontally stretched to a period of 4pi
Vertical shift of -1
A phase shift of pi units left
Where does the new function:
4a. cross the y-axis
4b. what is the maximum?
4c. What is the minimum?
HELP ASAP please
Answers
ANSWER:
1)
The function
f(x)=x+e−x
has a critical point (local minimum) at c=0.
The derivative is zero at this point.
f(x)=x+e−x.f′(x)=(x+e−x)′=1−e−x.f′(c)=0,⇒1−e−c=0,⇒e−c=1,⇒e−c=e0,⇒c=0.
2)The function f(x)=|x−3|
has a critical point (local minimum) at c=3.
The derivative does not exist at this point.
A critical point x=c
is a local maximum if the function changes from increasing to decreasing at that point.
3)The function f(x)=2x−x2
has a critical point (local maximum) at c=1.
The derivative is zero at this point.f(x)=2x−x2.f′(x)=(2x−x2)′=2−2x.f′(c)=0,⇒2−2c=0,⇒c=1.
4)The function f(x)=1−|x+2|
has a critical point (local maximum) at c=−2.
The derivative does not exist at this point.
A critical point x=c is an inflection point if the function changes concavity at that point.
5)The function f(x)=x3 has a critical point (inflection point) at c=0.The first and second derivatives are zero at c=0.f(x)=x3.f′(x)=(x3)′=3x2.f′(c)=0,⇒3c2=0,⇒c=0.
6)Trivial case: Each point of a constant function is critical. For example, any point c>0
of the function f(x)={2−x,x≤02,x>0 is acriticalpoint since f′(c)=0.f(x)={2−x,x≤02,x>0.f′(x)={−1,x≤00,x>0.f′(c)=0,⇒c>0.