If $(b^{2} - 4 a c)^{2} (1 + 4 a^{2}) < 64 a^{2}, a < 0$ , then the maximum value of quadratic expression $a x^{2} + b x + c$ is always less than

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Solution

The correct option is D $2$
As $a < 0$ so it is opened downward, Maximum value of given quadratic expression will be $- \frac{(b^{2} - 4 a c)}{4 a}$ , therefore after rearranging the given condition we have ( $\frac{b^{2} - 4 a c)^{2}}{({4 a)}^{2}} < 4$
Taking square root then we will get the max value which is $2.$
Hence, option $B$ is correct.

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