First slide

Theory of expressions

Question

Let $a, b, c$ be non-zero real number such that
$\begin{array}{l} \int_{0}^{1} (1 + \cos^{8} x) (a x^{2} + b x + c) d x \\ = \int_{0}^{2} (1 + \cos^{8} x) (a x^{2} + b x + c) d x \end{array}$
Then the quadratic equation $a x^{2} + b x + c = 0$ has

Moderate

A

no root in $(0, 2)$
B

at least one root in $(1, 2)$
C

double root $(0, 2)$
D

none of these

Solution

Let $f (β) < 0$ $f (x) = (1 + \cos^{8} x) (a x^{2} + b x + c)$
We are given
$\begin{array}{l} \int_{0}^{1} f (x) d x = \int_{0}^{2} f (x) d x \\ \int_{0}^{1} f (x) d x = \int_{0}^{1} f (x) d x + \int_{1}^{2} f (x) d x \\ \Rightarrow \int_{1}^{2} f (x) d x = 0 \end{array}$
If $f (x) > 0 (f (x) < 0) \forall x \in [1, 2]$
then $\int_{1}^{2} f (x) d x > 0 (\int_{1}^{2} f (x) d x < 0)$
But $\int_{1}^{2} f (x) d x = 0$
$∴ f (x)$ is partly positive and partly negative on $[1, 2]$ .
$\Rightarrow$ there exist $α, β \in [1, 2]$ such that
$f (α) > 0$ and $f (β) < 0$ .
As $f$ is continuous on [1,2] there exists $γ$ lying between $α$ and $β$ (and hence between 1 and 2) such that $f (γ) = 0$
$\Rightarrow (1 + \cos^{8} γ) (a γ^{2} + b γ + c) = 0$
$\Rightarrow a γ^{2} + b γ + c = 0 [∵ 1 + \cos^{8} γ \geq 1]$
Thus, $a x^{2} + b x + c = 0$ has at least one root in $[1, 2]$ .

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