Let $A$ be the set of all functions $f:mathbb{R}tomathbb{R}$ that satisfy the following two properties.
$begingroup$
Let $A$ be the set of all functions $f:mathbb{R}tomathbb{R}$ that satisfy the following two properties:
(1) $f$ has derivative of all orders, and
(2) for all $x,y in mathbb{R}$, $f(x+y)-f(y-x)=2xf'(y)$.
Which of the following sentences is true?
(a) Any $fin A$ is a polynomial of degree less than or equal to 1
(b) Any $f in A$ is a polynomial of degree less than or equal to 2
(c) $exists f in A$ which is not polynomial
(d) $exists f in A$ which is a polynomial of degree 4
It is a problem from TIFR GS-2018. I have found polynomials upto degree 2 hold this properties but have to exclude the possibility of (c) and (d). I also found out if $f'(x)$ has at most one real root and clearly graph of $f$ is symetric about that root. How can I proceed further?
calculus real-analysis
asked Nov 26 '18 at 3:47
OfflawOfflaw
2689
$endgroup$
add a comment |
$begingroup$
Let $A$ be the set of all functions $f:mathbb{R}tomathbb{R}$ that satisfy the following two properties:
(1) $f$ has derivative of all orders, and
(2) for all $x,y in mathbb{R}$, $f(x+y)-f(y-x)=2xf'(y)$.
Which of the following sentences is true?
(a) Any $fin A$ is a polynomial of degree less than or equal to 1
(b) Any $f in A$ is a polynomial of degree less than or equal to 2
(c) $exists f in A$ which is not polynomial
(d) $exists f in A$ which is a polynomial of degree 4
It is a problem from TIFR GS-2018. I have found polynomials upto degree 2 hold this properties but have to exclude the possibility of (c) and (d). I also found out if $f'(x)$ has at most one real root and clearly graph of $f$ is symetric about that root. How can I proceed further?
calculus real-analysis
asked Nov 26 '18 at 3:47
OfflawOfflaw
2689
$endgroup$
add a comment |
$begingroup$
Let $A$ be the set of all functions $f:mathbb{R}tomathbb{R}$ that satisfy the following two properties:
(1) $f$ has derivative of all orders, and
(2) for all $x,y in mathbb{R}$, $f(x+y)-f(y-x)=2xf'(y)$.
Which of the following sentences is true?
(a) Any $fin A$ is a polynomial of degree less than or equal to 1
(b) Any $f in A$ is a polynomial of degree less than or equal to 2
(c) $exists f in A$ which is not polynomial
(d) $exists f in A$ which is a polynomial of degree 4
It is a problem from TIFR GS-2018. I have found polynomials upto degree 2 hold this properties but have to exclude the possibility of (c) and (d). I also found out if $f'(x)$ has at most one real root and clearly graph of $f$ is symetric about that root. How can I proceed further?
calculus real-analysis
asked Nov 26 '18 at 3:47
OfflawOfflaw
2689
$endgroup$
Let $A$ be the set of all functions $f:mathbb{R}tomathbb{R}$ that satisfy the following two properties:
(1) $f$ has derivative of all orders, and
(2) for all $x,y in mathbb{R}$, $f(x+y)-f(y-x)=2xf'(y)$.
Which of the following sentences is true?
(a) Any $fin A$ is a polynomial of degree less than or equal to 1
(b) Any $f in A$ is a polynomial of degree less than or equal to 2
(c) $exists f in A$ which is not polynomial
(d) $exists f in A$ which is a polynomial of degree 4
It is a problem from TIFR GS-2018. I have found polynomials upto degree 2 hold this properties but have to exclude the possibility of (c) and (d). I also found out if $f'(x)$ has at most one real root and clearly graph of $f$ is symetric about that root. How can I proceed further?
calculus real-analysis
calculus real-analysis
asked Nov 26 '18 at 3:47
OfflawOfflaw
2689
asked Nov 26 '18 at 3:47
OfflawOfflaw
2689
asked Nov 26 '18 at 3:47
OfflawOfflaw
2689
asked Nov 26 '18 at 3:47
OfflawOfflaw
2689
asked Nov 26 '18 at 3:47
OfflawOfflaw
2689
2689
add a comment |
add a comment |
1 Answer
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$begingroup$
We have
$$f(x+y)-f(y-x)=2xf'(y) $$
Apply $frac{mathrm d}{mathrm dx}$:
$$f'(x+y)+f'(y-x)=2f'(y) $$
and once more:
$$f''(x+y)-f''(y-x)=0. $$
With $xleftarrow frac t2$, $tleftarrow frac t2$, this becomes
$$f''(t)=f''(0). $$
Thus $f$ is a polynomal and of degree $le 2$.
answered Nov 26 '18 at 4:08
Hagen von EitzenHagen von Eitzen
277k22269496
$endgroup$
$begingroup$
Thanks. But, there may be a typing error, it should $y leftarrow frac{t}{2}$
$endgroup$
– Offlaw
Nov 26 '18 at 5:21
add a comment |
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1 Answer
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1 Answer
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$begingroup$
We have
$$f(x+y)-f(y-x)=2xf'(y) $$
Apply $frac{mathrm d}{mathrm dx}$:
$$f'(x+y)+f'(y-x)=2f'(y) $$
and once more:
$$f''(x+y)-f''(y-x)=0. $$
With $xleftarrow frac t2$, $tleftarrow frac t2$, this becomes
$$f''(t)=f''(0). $$
Thus $f$ is a polynomal and of degree $le 2$.
answered Nov 26 '18 at 4:08
Hagen von EitzenHagen von Eitzen
277k22269496
$endgroup$
$begingroup$
Thanks. But, there may be a typing error, it should $y leftarrow frac{t}{2}$
$endgroup$
– Offlaw
Nov 26 '18 at 5:21
add a comment |
$begingroup$
We have
$$f(x+y)-f(y-x)=2xf'(y) $$
Apply $frac{mathrm d}{mathrm dx}$:
$$f'(x+y)+f'(y-x)=2f'(y) $$
and once more:
$$f''(x+y)-f''(y-x)=0. $$
With $xleftarrow frac t2$, $tleftarrow frac t2$, this becomes
$$f''(t)=f''(0). $$
Thus $f$ is a polynomal and of degree $le 2$.
answered Nov 26 '18 at 4:08
Hagen von EitzenHagen von Eitzen
277k22269496
$endgroup$
$begingroup$
Thanks. But, there may be a typing error, it should $y leftarrow frac{t}{2}$
$endgroup$
– Offlaw
Nov 26 '18 at 5:21
add a comment |
$begingroup$
We have
$$f(x+y)-f(y-x)=2xf'(y) $$
Apply $frac{mathrm d}{mathrm dx}$:
$$f'(x+y)+f'(y-x)=2f'(y) $$
and once more:
$$f''(x+y)-f''(y-x)=0. $$
With $xleftarrow frac t2$, $tleftarrow frac t2$, this becomes
$$f''(t)=f''(0). $$
Thus $f$ is a polynomal and of degree $le 2$.
answered Nov 26 '18 at 4:08
Hagen von EitzenHagen von Eitzen
277k22269496
$endgroup$
We have
$$f(x+y)-f(y-x)=2xf'(y) $$
Apply $frac{mathrm d}{mathrm dx}$:
$$f'(x+y)+f'(y-x)=2f'(y) $$
and once more:
$$f''(x+y)-f''(y-x)=0. $$
With $xleftarrow frac t2$, $tleftarrow frac t2$, this becomes
$$f''(t)=f''(0). $$
Thus $f$ is a polynomal and of degree $le 2$.
answered Nov 26 '18 at 4:08
Hagen von EitzenHagen von Eitzen
277k22269496
answered Nov 26 '18 at 4:08
Hagen von EitzenHagen von Eitzen
277k22269496
answered Nov 26 '18 at 4:08
Hagen von EitzenHagen von Eitzen
277k22269496
answered Nov 26 '18 at 4:08
Hagen von EitzenHagen von Eitzen
277k22269496
277k22269496
$begingroup$
Thanks. But, there may be a typing error, it should $y leftarrow frac{t}{2}$
$endgroup$
– Offlaw
Nov 26 '18 at 5:21
add a comment |
$begingroup$
Thanks. But, there may be a typing error, it should $y leftarrow frac{t}{2}$
$endgroup$
– Offlaw
Nov 26 '18 at 5:21
$begingroup$
Thanks. But, there may be a typing error, it should $y leftarrow frac{t}{2}$
$endgroup$
– Offlaw
Nov 26 '18 at 5:21
$begingroup$
Thanks. But, there may be a typing error, it should $y leftarrow frac{t}{2}$
$endgroup$
– Offlaw
Nov 26 '18 at 5:21
add a comment |
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