Question

Verify Rolle's theorem for the following function on the indicated interval f(x) = x² − 5x + 4 on [1, 4] ?

Answer 1

According to Rolle’s theorem, if f(x) is a real valued function defined on [a, b] such that it is continuous on [a, b], it is differentiable on (a, b) and f(a) = f(b), then there exists a real number c ∈(a, b) such that f(c) = 0.

Now, f(x) is defined for all x ∈[1, 4].
At each point of [1, 4], the limit of f(x) is equal to the value of the function. Therefore, f(x) is continuous on [1, 4].

Also,f' (x) = 2x - 5 exists for all x ∈ (1, 4).

So, f(x) is differentiable on (1, 4).

Also,
 f(1) = f(4) = 0

Thus, all the three conditions of Rolle’s theorem are satisfied.

Now, we have to show that there exists c ∈(1, 4) such that f'(c) = 0.

We have

f' (x) = 2x - 5

\[\therefore f'\left( x \right) = 0\] 

\[ \Rightarrow 2x - 5 = 0\] 

\[ \Rightarrow x = \frac{5}{2}\]

[Since `c = 5/2` ∈ (1, 4) such that f' (c) = 0 ]

Hence, Rolle’s theorem is verified.