Find all values of $k$ for which the inequality , $2 x^{2} - 4 k^{2} x - k^{2} + 1 > 0$ is valid for all real $x$ which do not exceed unity in the absolute value.

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Solution

As we know that if $a x^{2} + b x + c > 0$
In only that case when $b^{2} - 4 a c < 0$
So, for this inequality :- $(- 4 k^{2})^{2} - 4 \times 2 \times (- k^{2} + 1) < 0$
=> $16 k^{4} + 8 k^{2} - 8 < 0$
=> $8 (2 k^{2} - 1) (k^{2} + 1) < 0$
As we know that $8 (k^{2} + 1) > 0$ for all value of k
so, $2 k^{2} - 1 < 0 => k^{2} < \frac{1}{2}$
So, $k \in (- \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$

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