Problems on Fundamental Theorem
Problems on Fundamental Theorem:
1. Verify Rolle's Theorem for the functions f (x) = x2 - 4x 8 in the intervals [1,3].
Solutions: f (x) = x2 - 4x 8 is continuous in [1,3] and f' (x) = 2x - 4 exists for all value in (1,3)
Hence all three conditions of the theorem are satiesfied.
Now consider f ' (c) = 0
i.e 2c - 4 = 0 ⇒ 2c = 4.
and hence Rolle's Theorem is varified.
2. Varify Rolle's Theorem for the functions
Therefore f' (x) exists for all x. Also,
Hence all the three conditions of the theorem are satiesfied.
Now consider f' (c) = 0
Hence three exists - 2 ∈ ( -3,0) such that
f' (-2) = 0
Hence Rolle's Theorem varified.