Written Homework 6
Math 1200
Instructions: Submit your work on Gradescope bySunday, September 29, 2023 at 11:59
pm. You must show all work.
1.SupposeB(x) is a continuous function whose derivative and second derivative are given
by
B
′
(x) = (x−13)
2
(x+ 4)
3/7
andB
′′
(x) =
17(x−13)(x+ 1)
7(x+ 4)
4/7
(a) Find the critical points ofB(x).
(b) Find all intervals whereB(x) is increasing.
(c) Find the location of the local extrema ofB(x).
(d) Find the critical points ofB
′
(x).
(e) Find all intervals whereB(x) is concave down.
(f) Find the inflection points ofB(x).
1
2. The graph of the derivative of a continuous functionq(x) is pictured below.
(a) Where isq(x) decreasing?
(b) Where isq(x) concave up?
(c) Where isq
′′
(x)<0?
(d) Find the location of the absolute extrema ofq(x) on the interval [5,8].
(e) Find the location of the absolute extrema ofq
′
(x) on the interval [5,8].
(f) What are the critical points ofq(x)?
(g) Find and classify the local extrema ofq(x).
(h) Find the inflection points ofq(x).
2
3.For each function below, determine thexandycoordinates of the absolute extrema (if
they exist) on the given interval. For each extremum found, state if it is an absolute
maximum or absolute minimum.
(a)f(x) =x−
3
√
xon [−1,4]
(b)f(x) =x+
1
x
on [0.2,4]
(c)f(x) =x
4
−8x
3
+ 16x
2
on [−1,3]
3
4. Letf(x) = 2x
4
−3x
2
+ 6.
(a) Find the tangent line off(x) atx= 2.
(b) Computef
′′
(x) andf
′′′
(x).
(c) Find the tangent line tof
′′
(x) atx= 1.
4
5. Solve the below problem:
5
6.Sketch the graph of a continuous functiongon the interval [−4,4] satisfying the given
properties.
•g
′
(x) = 0 atx=−2 andx= 3
•g
′
(1) is undefined
•ghas an absolute minimum atx= 1
•ghas neither a local maximum nor a local minimum atx= 3
•ghas an absolute minimum atx=−2
−5−4−3−2−112345
−6
−5
−4
−3
−2
−1
1
2
3
4
5
6
x
y
6
