JEE Advance - Mathematics (2004 - No. 12)
Using Rolle's theorem, prove that there is at least one root
in $$\left( {{{45}^{1/100}},46} \right)$$ of the polynomial
$$P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035$$.
in $$\left( {{{45}^{1/100}},46} \right)$$ of the polynomial
$$P\left( x \right) = 51{x^{101}} - 2323{\left( x \right)^{100}} - 45x + 1035$$.
Rolle's theorem cannot be applied to this polynomial on the given interval.
P(45^(1/100)) = P(46), hence by Rolle's theorem, there exists at least one root in (45^(1/100), 46).
P(45^(1/100)) = P(46) = 0, hence by Rolle's theorem, there exists at least one root in (45^(1/100), 46).
P(45^(1/100)) != P(46), hence Rolle's theorem does not guarantee a root in the interval.
P(x) is not differentiable in (45^(1/100), 46), so Rolle's theorem is not applicable.
Comments (0)
