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The Mean Value Theorem- HW Problems
Determine if Rolle’s theorem applies to 𝑓(𝑥) on the given interval. If it
does, find all values 𝑐 that satisfy the conclusion of Rolle’s theorem. If
it doesn’t satisfy Rolle’s theorem, explain why it doesn’t.
1. 𝑓(𝑥 ) = 𝑥 2 − 4𝑥 𝑜𝑛 [0,4]
1
2. 𝑓(𝑥 ) = √𝑥 − 𝑥 𝑜𝑛 [0,4]
2
2
3. 𝑓(𝑥 ) = 𝑥 − 3
3 𝑜𝑛 [−8.8]
4. 𝑓(𝑥 ) = cos(𝑥 ) 𝑜𝑛 [0,2𝜋]
5. 𝑓(𝑥 ) = tan(𝑥 ) 𝑜𝑛 [0. 𝜋]
Determine if the mean value theorem applies to 𝑓(𝑥) in the given
interval. If it does, find all values 𝑐 that satisfy the conclusion of the
mean value theorem. If it doesn’t satisfy the MVT, explain why it
doesn’t.
6. 𝑓(𝑥 ) = 2𝑥 3 − 6𝑥 + 2 𝑜𝑛 [0,2]
7. 𝑓(𝑥 ) = |𝑥 | 𝑜𝑛 [−2,3]
2
8. 𝑓(𝑥 ) = 𝑥 3 𝑜𝑛 [0,8]
9. 𝑓(𝑥 ) = 2𝑠𝑖𝑛(𝑥 ) + sin(2𝑥 ) 𝑜𝑛 [0, 𝜋]
10. Show that the equation 𝑥 3 − 27𝑥 + 𝐾 = 0 has at most one root
in [−2,2] (a root is a point 𝑐 such that 𝑐 3 − 27𝑐 + 𝐾 = 0). Hint:
suppose 𝑓(𝑥 ) = 𝑥 3 − 27𝑥 + 𝐾 has two roots in [−2,2], 𝑐1 and 𝑐2 .
Now apply Rolle’s theorem to [𝑐1 , 𝑐2 ].
The Mean Value Theorem- HW Problems
Determine if Rolle’s theorem applies to 𝑓(𝑥) on the given interval. If it
does, find all values 𝑐 that satisfy the conclusion of Rolle’s theorem. If
it doesn’t satisfy Rolle’s theorem, explain why it doesn’t.
1. 𝑓(𝑥 ) = 𝑥 2 − 4𝑥 𝑜𝑛 [0,4]
1
2. 𝑓(𝑥 ) = √𝑥 − 𝑥 𝑜𝑛 [0,4]
2
2
3. 𝑓(𝑥 ) = 𝑥 − 3
3 𝑜𝑛 [−8.8]
4. 𝑓(𝑥 ) = cos(𝑥 ) 𝑜𝑛 [0,2𝜋]
5. 𝑓(𝑥 ) = tan(𝑥 ) 𝑜𝑛 [0. 𝜋]
Determine if the mean value theorem applies to 𝑓(𝑥) in the given
interval. If it does, find all values 𝑐 that satisfy the conclusion of the
mean value theorem. If it doesn’t satisfy the MVT, explain why it
doesn’t.
6. 𝑓(𝑥 ) = 2𝑥 3 − 6𝑥 + 2 𝑜𝑛 [0,2]
7. 𝑓(𝑥 ) = |𝑥 | 𝑜𝑛 [−2,3]
2
8. 𝑓(𝑥 ) = 𝑥 3 𝑜𝑛 [0,8]
9. 𝑓(𝑥 ) = 2𝑠𝑖𝑛(𝑥 ) + sin(2𝑥 ) 𝑜𝑛 [0, 𝜋]
10. Show that the equation 𝑥 3 − 27𝑥 + 𝐾 = 0 has at most one root
in [−2,2] (a root is a point 𝑐 such that 𝑐 3 − 27𝑐 + 𝐾 = 0). Hint:
suppose 𝑓(𝑥 ) = 𝑥 3 − 27𝑥 + 𝐾 has two roots in [−2,2], 𝑐1 and 𝑐2 .
Now apply Rolle’s theorem to [𝑐1 , 𝑐2 ].
