1
Derivatives and the Shapes of Graphs- HW Problems
For the following functions determine where the graph is
increasing/decreasing, concave up/down, and identify all local
maxima/minima and inflection points.
1. 𝑓(𝑥 ) = 𝑥 3 − 6𝑥 2 + 12𝑥
2. 𝑓(𝑥 ) = 𝑥 4 − 2𝑥 2 + 1
𝑥
3. 𝑔(𝑥 ) =
𝑥2 +1
4. ℎ(𝑥 ) = sin(𝑥 ) + cos(𝑥 ) on [0, 2𝜋]
5. 𝑔(𝑥 ) = 𝑥 √𝑥 + 1
Use the second derivative test to determine all relative maxima and
minima.
6. 𝑓(𝑥 ) = 𝑥 4 − 4𝑥 3 + 1
7. 𝑔(𝑥 ) = 𝑥 4 − 8𝑥 2
𝑥2
8. 𝑔(𝑥 ) =
𝑥−1
𝜋 3𝜋
9. ℎ(𝑥 ) = cos2 (𝑥 ) on (− , 2)
2
Derivatives and the Shapes of Graphs- HW Problems
For the following functions determine where the graph is
increasing/decreasing, concave up/down, and identify all local
maxima/minima and inflection points.
1. 𝑓(𝑥 ) = 𝑥 3 − 6𝑥 2 + 12𝑥
2. 𝑓(𝑥 ) = 𝑥 4 − 2𝑥 2 + 1
𝑥
3. 𝑔(𝑥 ) =
𝑥2 +1
4. ℎ(𝑥 ) = sin(𝑥 ) + cos(𝑥 ) on [0, 2𝜋]
5. 𝑔(𝑥 ) = 𝑥 √𝑥 + 1
Use the second derivative test to determine all relative maxima and
minima.
6. 𝑓(𝑥 ) = 𝑥 4 − 4𝑥 3 + 1
7. 𝑔(𝑥 ) = 𝑥 4 − 8𝑥 2
𝑥2
8. 𝑔(𝑥 ) =
𝑥−1
𝜋 3𝜋
9. ℎ(𝑥 ) = cos2 (𝑥 ) on (− , 2)
2
