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
Solutions:
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.