Grima MAT 212 - Chapter 3 in class practice problems part 1
1) Consider the graph of the function f(x) below.
a) the open intervals where the function graphed
is increasing
b) the open intervals where the function graph is
decreasing
c) the coordinates of all relative maxima
d) the coordinates of all relative minima
2) Consider the graph of the function f(x) below.
a) Find the open intervals where the function is
concave upward
b) Find the open intervals where the function is
concave downward
c) Find all inflection points
3) 𝑓(𝑥) = 2𝑥 3 − 6𝑥 2 − 8
a) f’(x)
b) the critical numbers
c) the open interval(s) where the function is increasing
d) the open interval(s) where the function is decreasing
e) the coordinates of all relative maxima
f) the coordinates of all relative minima
4) 𝑓(𝑥) = 4𝑥𝑒 𝑥
a) f’(x)
b) the critical numbers
c) the open interval(s) where the function is increasing
d) the open interval(s) where the function is decreasing
e) the coordinates of all relative maxima
f) the coordinates of all relative minima
5) 𝑓(𝑥) = 𝑥 4 − 6𝑥 2
a) Find the open intervals where the function is concave upward.
b) Find the open intervals where the function is concave downward.
c) Find all inflection points
Answers ?????
1a) (−∞, −6) ∪ (1, ∞)
1b) (-6,1)
1c) relative maximum of y = 32.4 which occurs at x = -6
1d) relative minimum of y = -1.9 which occurs at x = 1
2a) (−2.5, ∞)
2b) (−∞, 2.5)
2
3a) f’(x) = 6x – 12x
3b) x = 0, 2
3c) (−∞, 0) ∪ (2, ∞)
3d) (0,2)
3e) relative maximum of y = -8 which occurs at x = 0
3f) relative minimum of y = -16 which occurs at x = 2
4a) f’(x) = 4ex(x+1)
4b) x = -1
4c) (−1, ∞)
4d) (−∞, −1)
4e) no relative maximum
4f) relative minimum of y =
5a) (−∞, −1) ∪ (1, ∞)
5b) (-1,1)
−4
𝑒
which occurs at x = -1
5c) (-1,-5) and (1, -5)
Grima MAT 212 - Chapter 3 in class practice problems part 2
6) f(x) = x3- 6x2
a) Find the domain
b) Find the x-intercept(s), if any
c) Find the y-intercept, in there is one
d) Find the interval(s) where the graph of the function is increasing
e) Find the interval(s) where the graph of the function is decreasing
f) Find all relative maxima and relative minima
g) Find the interval(s) where the graph of the function is concave up (if any)
h) Find the interval(s) where the graph of the function is concave down (if any)
i) Find all inflection points (if any)
j) Sketch a graph
7) f(x) = 7xex (hint f”(x) = ex(7x+14)
a) Find the domain
b) Find the x-intercept(s), if any
c) Find the y-intercept, in there is one
d) Find the interval(s) where the graph of the function is increasing
e) Find the interval(s) where the graph of the function is decreasing
f) Find all relative maxima and relative minima
g) Find the interval(s) where the graph of the function is concave up (if any)
h) Find the interval(s) where the graph of the function is concave down (if any)
i) Find all inflection points (if any)
j) Sketch a graph
8) Find the following. 𝑓(𝑥) =
2𝑥+10
𝑥−2
a) Find the domain and vertical asymptotes
b) Find the x-intercept(s), if any
c) Find the y-intercept, in there is one
d) Find all horizontal asymptotes
−14
e) Find the interval(s) where the graph of the function is increasing (hint: 𝑓 ′ (𝑥) = (𝑥−2)2)
f) Find the interval(s) where the graph of the function is decreasing
g) Find all relative maxima and relative minima
28
h) Find the interval(s) where the graph of the function is concave up (if any) (hint: 𝑓"(𝑥) = (𝑥−2)3
i) Find the interval(s) where the graph of the function is concave down (if any)
j) Find all inflection points (if any)
k) Sketch a graph
Answers ?????
6a) (−∞, ∞) 6b) (0,0) (6,0)
6c) (0,0)
6d) (−∞, 0) ∪ (4, ∞)
6e) (0,4)
6f) relative maxima of y = 0 occurs at x = 0
relative minima of y =-32 which occurs at x = 4
6g) (2, ∞)
6h) (−∞, 2) 6i) (2,-16)
6j)
7a) (−∞, ∞)
7b) (0,0)
7f) no relative maximum
7g) (−2, ∞)
relative
7h) (−∞, −2)
7c) (0,0)
7d) (−1, ∞)
−7
minimum of y = 𝑒 which
−14
7i) (−2, 2 )
𝑒
7e) (−∞, −1)
occurs at x = -1
7j)
8a) VA x = 2, domain (−∞, 2) ∪ (2, ∞)
8d) y = 2
8e) never
8g) no relative maxima nor minima
8j) no inflection points
8k)
8b) (-5,0)
8f) (−∞, 2) ∪ (2, ∞)
8h) (2, ∞)
8c) (0,-5)
8i) (−∞, 2)