A) \[[{{\log }_{e}}2,\infty )\]
B) \[[{{\log }_{e}}3,\infty )\]
C) \[[2{{\log }_{e}}3,\infty )\]
D) \[[0,\infty )\]
E) \[[2{{\log }_{e}}2,\infty )\]
Correct Answer: E
Solution :
Function\[f(x)={{\log }_{e}}(3{{x}^{2}}+4)\] Let \[y={{\log }_{e}}(3{{x}^{2}}+4)\] \[\Rightarrow \] \[3{{x}^{2}}+4={{e}^{y}}\] \[x=\sqrt{\frac{{{e}^{y}}-4}{3}}\] Here, \[\frac{{{e}^{y}}-4}{3}\ge 0\] \[\Rightarrow \] \[{{e}^{y}}\ge 4\] \[\Rightarrow \] \[y\ge 2{{\log }_{e}}2\] Hence, the range of the function is\[[2{{\log }_{e}}2,\infty )\].
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