Verify Rolle's theorem in the interval $[1, 2]$ for $f (x) = x^{2} - 3 x + 4$ .

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Solution

$(a) f (x) i s a p o l y n o m i a l f u n c t i o n a n d h e n c e c o n t i n u o u s i n [1, 2] (b) f (x) i s a p o l y n o m i a l f u n c t i o n a n d h e n c e d i f f e r e n t i a b l e i n (1, 2) t h e n f^{'} (c) = (\frac{f (2) - f (1)}{2 - 1}) 2 c - 3 = (2 - 2) / 1 ∴ c = (\frac{3}{2}) \in [1, 2] H e n c e v e r i f i e d$

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