Question

State Rolle’s theorem and also verify Rolle’s theorem for the function
x2=5x+4 on [1,4]

Answer 1

Rolle's theorem is a special case of the mean value theorem. Rolle's theorem states that for any continuous, differentiable function that has two equal values at two distinct points, the function must have a point on the function where the first derivative is zero

f(x) = x^2 -5x -4 ,
for rolle’s theorem,
f(x) should be continuous in (a, b)
f(x) should be diffrentiable in (a, b)
if f(a)= f(b) then there exist a point c in(a, b) such that f’(c) = 0
here ,
f(1) = 1^2 -5×1 -4 = 1-5-4 = -8
f(4) = 4^2 -5×4-4 = 16-20-4 = 0

f(1) ≠f4) hence rolles theorem can’t be applied
Thanks