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What functions check these conditions?
TheBlackPuzzle913 2
N
by Filipjack
Source: RMO shortlist, Mihai Bunget and Dragoș Gabriel Borugă
Find all functions 
that are twice differentiable and satisfy 
that are twice differentiable and satisfy 
2 replies
Show the existence of a neighborhhod
Alidq 0
Source: some derivative problem handout
Let 
a twice differentiable function with a continuous second derivative, for which there exists a unique 
such that 
. If 
, show that there exists a neighborhood 
of 
such that for any 
, the sequence defined by the recurrence relation 
converges to .
a twice differentiable function with a continuous second derivative, for which there exists a unique 
such that 
. If 
, show that there exists a neighborhood 
of 
such that for any 
, the sequence defined by the recurrence relation 
converges to .
0 replies
f"(x)>0, show that int(f(x)cosx dx) >0
Sayan 11
N
by Mathzeus1024
Source: ISI(BS) 2009 #2
Let 
be a continuous function, whose first and second derivatives are continuous on ![$[0,2\pi]$](//latex.artofproblemsolving.com/1/6/6/166f2f81461a7a6ce563dbb02f64f78f6bd08acc.png)
and 
for all 
in . Show that
![\[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\]](//latex.artofproblemsolving.com/d/e/a/dea20971d99aa4120cd34c5817c179da5e642b43.png)
be a continuous function, whose first and second derivatives are continuous on ![$[0,2\pi]$](http://latex.artofproblemsolving.com/1/6/6/166f2f81461a7a6ce563dbb02f64f78f6bd08acc.png)
and 
for all 
in . Show that![\[\int_{0}^{2\pi} f(x)\cos x dx \geq 0\]](http://latex.artofproblemsolving.com/d/e/a/dea20971d99aa4120cd34c5817c179da5e642b43.png)
11 replies
Double derivative coated integral
RenheMiResembleRice 1
N
by HacheB2031
Source: Lufang Yue, Lianru Meng
Show that J>0. For this to be true, I guess we also need the condition for c and d to be positive, isn't that right?
1 reply
prove that there exists \xi
Peter 21
N
by Alphaamss
Source: IMC 1998 day 1 problem 4
The function 
is twice differentiable and satisfies 
.
Prove that there is a![$\xi \in ]0,1[$](//latex.artofproblemsolving.com/f/a/c/fac71dc4df50908a85c3133787a84ccb2ae3d79b.png)
for which we have 
.
is twice differentiable and satisfies 
.Prove that there is a
![$\xi \in ]0,1[$](http://latex.artofproblemsolving.com/f/a/c/fac71dc4df50908a85c3133787a84ccb2ae3d79b.png)
for which we have 
.
21 replies
Derivative of function R^2 to R^2
Sifan.C.Maths 1
N
by alexheinis
Source: Internet
Give a function 
. Calculate the first and second derivative of the function at the point 
.
. Calculate the first and second derivative of the function at the point 
.
1 reply
Derivative of Normalization Map has null space of dimension 1
myth17 4
N
by myth17
Let 
be defined on 
. Show that the dimension of the kernel of 
for any 
is 
.
be defined on 
. Show that the dimension of the kernel of 
for any 
is 
.
4 replies
if 
and 
.
Cycle length of (x^2-1)/2x derivative relation power of two crazy
ehz2701 2
N
by rchokler
Let
[center]
[/center]
(this is the approximation using Newton’s method for
. Consider the sequence given by 
. Suppose for some 
, we have the relationship 
, where 
for 
(in other words, a cycle of length 
). Show that
[center]
, [/center]
where
denotes function compositions of (i.e. 
, 
.)
—————
For example, when
, it produces a cycle of length 
, and 
[center]
[/center](this is the approximation using Newton’s method for
. Consider the sequence given by 
. Suppose for some 
, we have the relationship 
, where 
for 
(in other words, a cycle of length 
). Show that[center]
, [/center]where
denotes function compositions of (i.e. 
, 
.)—————
For example, when
, it produces a cycle of length 
, and 
2 replies
Differentiable function
kido2006 1
N
by alexheinis
Let ![$ f:\left [ 0,+\infty \right ] \rightarrow \mathbb{R}$](//latex.artofproblemsolving.com/5/b/7/5b70e424d4f40dacd6e70370fafb1a2cf2f519ba.png)
be a twice differentiable function satisfying:
Prove that there exists 
such that 
![$ f:\left [ 0,+\infty \right ] \rightarrow \mathbb{R}$](http://latex.artofproblemsolving.com/5/b/7/5b70e424d4f40dacd6e70370fafb1a2cf2f519ba.png)
be a twice differentiable function satisfying:
Prove that there exists 
such that 
1 reply
