Rolle's and mean value theorems

These two theorems were removed from the rationalised NCERT chapter but are asked in JEE. Both need f continuous on [a, b] and differentiable on (a, b).

Rolle's theorem: if also f(a) = f(b), there is at least one c in (a, b) with f′(c) = 0, a point where the tangent is horizontal.

Worked example: f(x) = x² + 2 on [−2, 2] has f(−2) = f(2) = 6, and f′(x) = 2x is 0 at c = 0, which lies inside the interval.

Mean value theorem: there is at least one c in (a, b) with f′(c) = [f(b) − f(a)]/(b − a). The tangent at c is parallel to the chord joining the ends.

Worked example: f(x) = x² on [2, 4]. The chord slope is (16 − 4)/2 = 6, and 2c = 6 gives c = 3, inside (2, 4).

Check the conditions before applying either theorem. |x| on [−1, 1] has equal end values but no c with f′(c) = 0, because it is not differentiable at 0.

Rolle's theorem is the special case of the mean value theorem where the chord is horizontal.