First slide

Trigonometric transformations

Question

Let $f (x) = x^{2} - 2 \sqrt{(\sin \sqrt{3} - \sin \sqrt{2})} x - (\cos \sqrt{3} - \cos \sqrt{2})$ then

Moderate

A

$f (x) is positive \forall x \in R$
B

f(x) assumes both positive and negative values
C

f(x)= 0 has no real roots
D

y = f(x) touches the line y = 0

Solution

We have
$\begin{array}{l} f (x) = x^{2} - 2 \sqrt{(\sin \sqrt{3} - \sin \sqrt{2})} x - (\cos \sqrt{3} - \cos \sqrt{2}) \\ ∴ D = 4 [(\sin \sqrt{3} - \sin \sqrt{2}) + (\cos \sqrt{3} - \cos \sqrt{2})] \\ = 8 \sqrt{2} \sin (\frac{\sqrt{3} - \sqrt{2}}{2}) \cos (\frac{π}{4} + \frac{\sqrt{2} + \sqrt{3}}{2}) \end{array}$
As $\sin (\frac{\sqrt{3} - \sqrt{2}}{2}) > 0 and \frac{π}{2} < \frac{\sqrt{2} + \sqrt{3}}{2} < π$
$∴ D_{1} < 0$
Hence, $f (x) > 0 \forall x \in R$

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