A) \[m{{e}^{mx}}({{m}^{2}}{{x}^{2}}+6mx+6)\]
B) \[2{{m}^{3}}x{{e}^{mx}}\]
C) \[m{{e}^{mx}}({{m}^{2}}{{x}^{2}}+2mx+2)\]
D) None of these
Correct Answer: A
Solution :
\[y={{x}^{2}}{{e}^{mx}}\] Differentiating w.r.t. \[x\], we get \[\frac{dy}{dx}=2x{{e}^{mx}}+m{{x}^{2}}{{e}^{mx}}\] Again, \[\frac{{{d}^{2}}y}{d{{x}^{2}}}=2({{e}^{mx}}+mx{{e}^{mx}})+m(2x{{e}^{mx}}+{{x}^{2}}m{{e}^{mx}})\] or \[\frac{{{d}^{2}}y}{d{{x}^{2}}}={{e}^{mx}}({{m}^{2}}{{x}^{2}}+4mx+2)\] Again, \[\frac{{{d}^{3}}y}{d{{x}^{3}}}={{e}^{mx}}[{{m}^{3}}{{x}^{2}}+4{{m}^{2}}x+2m+2{{m}^{2}}x+4m]\] \[={{e}^{mx}}[{{m}^{3}}{{x}^{2}}+6{{m}^{2}}x+6m]\] \[\Rightarrow \frac{{{d}^{3}}y}{d{{x}^{3}}}=m{{e}^{mx}}({{m}^{2}}{{x}^{2}}+6mx+6)\].
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