Question

state rolle' thoeorem and also verify rolle's theorem for the function x2=5x+4 on [1,4]

Answer 1

Solution:-

given by:-
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
I tink question is posted wrong


■I HOPE ITS HELP■

Answer 2

Rolle's theorem states that,
Let  f(x) be any continuous function then,

i)It is continuous in closed interval [a,b]
ii)It is differentiable in open interval (a,b)
iii)If f(a)=f(b)
 then there exist at least one point c such that c∈(a,b) and f'(c)=0

Now, in the question,
a=1, b= 4
So,
f(x)=x² -5x-4
f(1)=-8
f(4)=0
Since f(a)=f(b),
It does not satisfy this MVT
So, Rolle's Theorem is not applicable