I.S.I. (Indian Statistical Institute) B.Stat/B.Math Entrance Examination 2013. Subjective Problem no. 2.
Medium
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[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]Do you really need a hint? Try it first!
[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.22.4"]This problem simply ask for the range of the function defined by f(x)=$latex \frac {1}{x+2cosx}$ compute the derivative of the function = $latex \frac {2sinx-1}{(x+2cosx)^2}$
[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.22.4"]First extrema occurs at $latex x$= $latex \frac{\pi}{6}$ The first derivative is negetive in the interval [ 0, $latex \frac{\pi}{6}$] hence the function is decreasing in this interval f(0)=$latex \frac{1}{2}$ ; f($latex \frac{\pi}{6}$)=$latex \frac{1}{\frac{\pi}{6}+ {\sqrt{3}}}$
[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.22.4"]For x>$latex \frac{\pi}{6}$ the derivative becomes positive , and remain so upto x=$latex \frac{5\pi}{6}$ after which it becomes negative thus we have minima at x= $latex \frac{\pi}{6}$ and maxima at x= $latex \frac{5\pi}{6}$ f($latex \frac{5\pi}{6}$)= $latex \frac{1}{\frac{5\pi}{6}+\sqrt{3}}$ note that as $latex x\rightarrow \infty$the denominator of the function increases
[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.22.4"]Hence we can conclude that $latex f(x)\rightarrow0$ clearly x=$latex \frac{5\pi}{6}$ gives the global maxima so , the range is (0,$latex \frac{1}{\frac{5\pi}{6}+\sqrt{3}}$]
[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" min_height="12px" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px"]Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.
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