Polynomial Function

Calculus Level 4

If a non-zero polynomial f ( x ) f(x) satisfies the condition f ( 2 x ) = f ′ ( x ) f ′ ′ ( x ) f(2x) = f'(x) f''(x) , find the value of f ( 3 ) f(3) .


The answer is 12.

Hardik Gupta by Hardik Gupta , 22, India

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2 solutions

Hardik Gupta
May 12, 2016

Let degree of function = n n
therefore, D E G f ( 2 x ) DEG{ f(2x) } = n n
D E G f ′ ( x ) DEG{ f '(x) } = ( n − 1 ) (n-1)
D E G f ′ ′ ( x ) DEG{ f ''(x) } = ( n − 2 ) (n-2)
so,
n n = ( n − 1 ) (n-1) + ( n − 2 ) (n-2)
=> n = 3 n = 3
let f ( x ) f(x) = a x 3 ax^3 + b x 2 bx^2 + c x cx + d d
=> f ( 2 x ) f(2x) = 8 a x 3 8ax^3 + 4 b x 2 4bx^2 + 2 c x 2cx + d d
also, f ′ ( x ) f '(x) = 3 a x 2 3ax^2 + 2 b x 2bx + c c
and f ′ ′ ( x ) f ''(x) = 6 a x 6ax + 2 b 2b


as f ( 2 x ) f(2x) = f ′ ( x ) f '(x) . f ′ ′ ( x ) f ''(x)
=> 8 a x 3 8ax^3 + 4 b x 2 4bx^2 + 2 c x 2cx + d d = ( 3 a x 2 + 2 b x + c ) ( 6 a x + 2 b ) ) ({3ax^2} + {2bx} + c)({6ax} + {2b}))
by comparing the coeffs.
8 a 8a = 18 a 2 18a^2
=> a a = 4 9 \frac{4}{9}
also, b = c = d = 0 b=c=d = 0
therefore,
f ( x ) f(x) = 4 9 \frac{4}{9} x 3 x^3
therefore,
f ( 3 ) f(3) = 4 9 \frac{4}{9} * 3 3 3^3
=> f ( 3 ) f(3) = 12. 12.

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Use \implies for ⟹ \implies symbol.

Samara Simha Reddy - 5 years, 1 month ago

Note that the case of "n=0" has to be dealt with separately, since the degree of the 0 polynomial is not -1.

This is why your solution missed the crucial requirement that the polynomial is "non-zero".

Calvin Lin Staff - 5 years ago
展豪 張
May 13, 2016

Is f ( x ) = 0 f(x)=0 a polynomial too? If so, the answer could be 12 12 or 0 0

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Read question one more time..it's written that if f(x) be a non zero polynomial.

Hardik Gupta - 5 years ago

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I didn't see it when I answer it......

展豪 張 - 5 years ago

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The problem was edited after the "solution" was posted. Those who answered 0 have been marked correct.

In future, if you spot any errors with a problem, you can “report” it by selecting "report problem" in the menu. This will notify the problem creator who can fix the issues.

Calvin Lin Staff - 5 years ago

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@Calvin Lin – If I have problems with Latex, where I do report it? (Just kidding). I need the codes of Latex because the ones I know uses the $$$ sign. Where can I find helpful hints with latex codes. Thanks.

Hana Wehbi - 5 years ago

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@Hana Wehbi – We use \ ( \ )instead of $ $ to surround Latex. Otherwise, the "codes" are all the same.

You can hover your mouse over the Latex image to see the underlying code.

E.g.

Calvin Lin Staff - 5 years ago

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@Calvin Lin – Thanks, that was really helpful. I will give it a try.

Hana Wehbi - 5 years ago

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