#Weird functional equation

31 messages · Page 1 of 1 (latest)

teal meadowBOT
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barren pike
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We have:
f(x) - 2f(-x) = 3x^2 ∫(f(t)dt, -1, 1) - 6
Now, let's try replacing x by -x. Then we get:
-2f(x) + f(-x) = 3x^2 ∫(f(t)dt, -1, 1) - 6
Now you can solve for f(x).

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Yeah, nice! So, we have:
f(x) = 6 - 3x^2 ∫(f(t)dt, -1, 1)
Now, considering that integral is constant, f(x) must be a quadratic function. Moreover, notice that f(0) = 6 and f(x) is even.
So, try taking f(x) = ax^2 + 6. You can substitute it into the equation and use undetermined coefficients.

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Because f(x) needs to be even.

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And f(0) = 6.

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Well, of course. An even function can't have odd powers.

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No worries!

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Better to calculate the integral separately so you don't get confused.

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What will the value of the integral be?

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Right, nice. So, we get:
ax^2 + 6 = 6 - 3x^2 (2a/3 + 12)
6 can obviously be cancelled, which leads to:
a = -3(2a/3 + 12)

warm ravine
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@barren pike are you doing the method in which the intergral can wtiteen as some k

barren pike
minor steppeBOT
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@umbral horizon has given 1 rep to @barren pike

barren pike
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Nice, you're welcome!

warm ravine
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Some thing like this

rn_image_picker_lib_temp_b0d01905-a264-450c-8fbf-5e0073339ff5.jpg
barren pike
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Well, we also got that result.

warm ravine
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Sorry qs I wrote wrong

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Is this what you did.??

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So 2nd and 3rd options are correct

barren pike
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We should get f(x) = 6 - 12x^2. So, options B, C and D should be correct.

warm ravine
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Yup d also

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Waht method did you use

barren pike
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You can see my approach above.

warm ravine
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Are you by chance writing jee advance. .

barren pike
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JEE? I believe that's a kind of exam?

warm ravine
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Yeah. In India the most difficult exam for admission into iits

barren pike
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Hm, I see.

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Well, I'm not in India, so no.

barren pike
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Oh, interesting.