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February 2, 2020

Continuous Function: TIFR 2017 Problem 10

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Let us take a warm up quiz

[/et_pb_text][et_pb_code _builder_version="4.1"][h5p id="16"][/et_pb_code][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Understand the problem

[/et_pb_text][et_pb_text _builder_version="4.1" text_font="Raleway||||||||" background_color="#f4f4f4" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px" box_shadow_style="preset2"] There exist a non-negative continuous function f: [0,1] \longrightarrow \mathbb{R} such that \int_{0}^{1} f^{n} dx \longrightarrow 2 as \longrightarrow \infty (a) TRUE (b) FALSE

[/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="4.1" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="4.1"]

TIFR PROBLEM 10[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="4.1" open="off"]Continuous Function [/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="4.1" open="off"]EASY[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="4.1" open="off"]REAL ANALYSIS BY S.K MAPA[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Start with hints

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="4.1" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="4.1"]Rather I want to say that it is a comment on the question that here f^{n} does not mean f \circ f \circ . . . . . . \circ f (n times). Here f^{n}= f \bullet f . . . . . \bullet f (n times). Now do you want to think again with this disclosure?[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="4.1"]Make two cases Case 1:  0<f(x) \leq 1\ \forall x \in [0,1] [Observe that f is non negative function] Case 2:  f(x) > 1 for some x \in [0,1] Now prove for each case that \int_{0}^{1} f^{n} dx \nrightarrow 2 as n \to \infty . So, by the last statement you have guessed the validity of the statement . It is a false statement!! In the next two cases, we will basically prove two cases.[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="4.1"]Case 1:  If f(x) \leq 1\ \forall\ x \in [0,1] then f^{n}(x) \leq 1\ \forall\ x \in [0,1] So, \int_{0}^{1} f^{n} dx \leq\ \int_{0}^{1} 1 dx = 1 \implies \lim_{n \to \infty} \int_{0}^{1} f^{n} dx \leq 1 So, \int_{0}^{1} f^{n} dx \nrightarrow 2 as n \to \infty [/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="4.1"]Case 2:  Suppose f(x) > 1 for some  y \in [0,1] Now, as f(x) is continuous function We have f(x) > 1+\epsilon\ \forall\ x \in (y - \delta, y + \delta) for some \epsilon , \delta > 0 \implies f^{n}(x) > (1+ \epsilon)^{n} then we have \int_{0}^{1} f^{n}\ dx > \int_{y - \delta} ^{y + \delta} (1+ \epsilon)^{n}\ dx [If y \in [0,1]] or \int_{1 - \delta}^{1} (1+ \epsilon)^{n}\ dx [If y=1 ] or \int_{0}^{0 + \delta} (1+ \epsilon)^{n}\ dx [If y=0 ] In either case , \int_{0}^{1} f^{n}\ dx > (1 + \epsilon)^{n}\ \delta \to \infty   So,the statement is false.[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.27.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3"]

Similar Problems

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The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.

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