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Manipulation of inequalities

Given an inte...

Question

Given an interval $[a, b]$ that satisfies hypothesis of Rolle's theorem for the function $f (x) = x^{4} + x^{2} - 2.$ It is known that $a = - 1.$ Then the possible value of $b$ is :

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Solution

We factor the polynomial:
$x^{4} + x^{2} - 2 = (x^{2} + 2) (x^{2} - 1) = (x^{2} + 2) (x - 1) (x + 1) .$
It is now easy to see that the function has two zeros: $x_{1} = - 1 ($ coincides with the value of $a)$ and $x_{2} = 1.$
Since the function is a polynomial, it is everywhere continuous and differentiable. So this function satisfies Rolle's theorem on the interval $[- 1, 1] .$ Hence, $b = 1$

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Manipulation of inequalities

Standard XII Mathematics

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