My Math Forum CSAT Problem 30 (2015)

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 July 31st, 2017, 04:52 PM #1 Newbie   Joined: Jul 2017 From: Brazil Posts: 4 Thanks: 0 CSAT Problem 30 (2015) Problem 30 A continuous function f(x) satisfies the following. For all reals $\displaystyle x ≤ b, f(x) = a(x - b)^2 + c$ For all reals, $\displaystyle f(x) = \int_0^x{ \sqrt{4 - f(t)}}dt$ if $\displaystyle \int_0^6{f(x) dt} = \frac{q}{p}$, where p, q are relatively prime positive integers, find p + q. I'm stuck.
 July 31st, 2017, 08:02 PM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,671 Thanks: 2651 Math Focus: Mainly analysis and algebra The question seems somewhat strange in the definition of $f(x) \le b$ without any indication of the value of $b$, but one approach would be to use the Fundamental Theorem of Calculus on the second definition to gather some information on $a$, $b$ and $c$. Following further down that path, attempting to evaluate the first given integral by means of a trigonometric substitution may allow you to use the given integral result to simplify the result and introduce $p$ and $q$. Note that, for the first integral to be valid for all real $x$, we must have that $f(t) \le 4$ for all real $t$. This also constrains $a$, $b$ and $c$. Thanks from cachorroloucobr
 July 31st, 2017, 09:10 PM #3 Global Moderator   Joined: Dec 2006 Posts: 20,833 Thanks: 2161 In the final equation, cachorroloucobr, is the integration meant to be with respect to $t$ rather than $x$, or is it mistyped? Thanks from cachorroloucobr
 July 31st, 2017, 09:19 PM #4 Newbie   Joined: Jul 2017 From: Brazil Posts: 4 Thanks: 0 Yeah, I mistyped $\displaystyle \int_0^6{f(x) dx} = \frac{q}{p}$
 July 31st, 2017, 10:22 PM #5 Math Team     Joined: Jul 2011 From: Texas Posts: 2,982 Thanks: 1575 $\displaystyle f(x)=\int_0^x \sqrt{4-f(t)} \, dt \implies f(0)=0$ and $f'(x)=\sqrt{4-f(x)} \implies f(x) \le 4$ $f(x)=a(x-b)^2 + c \implies f(0)=ab^2+c$ and $f'(x)=2a(x-b)$ from that, $2a(x-b) = \sqrt{4-f(x)}$ note $x \le b \implies a \le 0$ $4a^2(x-b)^2 = 4-f(x) \implies f(x) = 4-4a^2(x-b)^2$ matching coefficients, $c=4$ and $a= -\dfrac{1}{4}$ finally, $ab^2+c=0 \implies -\dfrac{b^2}{4}+4 = 0 \implies b = \pm 4$ note $f'(0)=2 \implies b=4$ Now ... how to deal with a definite integral with an upper limit of integration, $x=6$ when $x \le 4$ ??? Thanks from cachorroloucobr
 July 31st, 2017, 11:58 PM #6 Global Moderator   Joined: Dec 2006 Posts: 20,833 Thanks: 2161 The value of $x$ can exceed 4.
August 1st, 2017, 05:45 AM   #7
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 Originally Posted by skeeter $\displaystyle f(x)=\int_0^x \sqrt{4-f(t)} \, dt \implies f(0)=0$ and $f'(x)=\sqrt{4-f(x)} \implies f(x) \le 4$ $f(x)=a(x-b)^2 + c \implies f(0)=ab^2+c$ and $f'(x)=2a(x-b)$ from that, $2a(x-b) = \sqrt{4-f(x)}$ note $x \le b \implies a \le 0$ $4a^2(x-b)^2 = 4-f(x) \implies f(x) = 4-4a^2(x-b)^2$ matching coefficients, $c=4$ and $a= -\dfrac{1}{4}$ finally, $ab^2+c=0 \implies -\dfrac{b^2}{4}+4 = 0 \implies b = \pm 4$ note $f'(0)=2 \implies b=4$ Now ... how to deal with a definite integral with an upper limit of integration, $x=6$ when $x \le 4$ ???
I don't understand:
Why do you assume that f (0) = 0?

August 1st, 2017, 06:29 AM   #8
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Quote:
 Originally Posted by cachorroloucobr I don't understand: Why do you assume that f (0) = 0?
no "assume" to it ...

recall a basic property of definite integrals ...

$\displaystyle \int_a^a g(x) \, dx = 0$

If $\displaystyle f(x) = \int_0^x \sqrt{4-f(t)} \, dt$, then $\displaystyle f(0) = \int_0^0 \sqrt{4-f(t)} \, dt = 0$

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