A) \[{{y}^{2}}.\log a{{b}^{2}}\]
B) \[y.\log a{{b}^{2}}\]
C) \[{{y}^{2}}\]
D) \[y.{{(\log {{a}^{2}}b)}^{2}}\]
E) \[y.{{(\log a{{b}^{2}})}^{2}}\]
Correct Answer: E
Solution :
\[y={{a}^{x}}{{b}^{2x-1}}\] \[\frac{dy}{dx}={{a}^{x}}{{b}^{2x-1}}\log a+2{{a}^{x}}{{b}^{2x-1}}\log b\] = \[{{a}^{x}}{{b}^{2x-1}}(\log a+2\log b)\] \[\frac{{{d}^{2}}y}{d{{x}^{2}}}={{a}^{x}}{{b}^{2x-1}}{{(\log a+2\log b)}^{2}}\] \[={{a}^{x}}{{b}^{2x-1}}{{(\log a{{b}^{2}})}^{2}}\]\[=y{{(\log a{{b}^{2}})}^{2}}\].
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