question_answer

Find    the    intervals    in    which \[f(x)=\frac{3}{10}{{x}^{4}}-\frac{4}{5}{{x}^{3}}-3{{x}^{2}}+\frac{36}{5}x+11\] is (a) Strictly increasing (b) strictiy decreasing.

Answer:

We have, \[f(x)=\frac{3}{10}{{x}^{4}}-\frac{4}{5}{{x}^{3}}-3{{x}^{2}}+\frac{36}{5}x+11\] \[f'(x)=\frac{12}{10}{{x}^{3}}-\frac{12}{5}{{x}^{2}}-6x+\frac{36}{5}=\frac{6}{5}(x-1)\,\,(x+2)\,\,(x-3)\]\[f(x)=0\Rightarrow x=1,\,\,-\,2,\,\,3\] \[\therefore \]Intervals are \[(-\,\infty ,\,\,-\,2),\]\[(-\,2,\,\,1)\]\[(1,\,\,3)\]and \[(3,\,\,\infty )\] \[\therefore \]\[f'(x)>0\]for \[(-\,2,\,\,1)\cup (3,\,\,\infty )\] \[f(x)\]is strictly increasing in\[(-\,2,\,\,1)\]and\[(3,\,\,\infty )\]. \[\therefore \]\[f'(x)<0\]for \[(-\,\infty ,\,\,-\,2)\]and \[(1,\,\,3)\] \[f(x)\]is strictly decreasing in \[(-\,\infty ,\,\,-\,2)\]and \[(1,\,\,3)\]