find the not null, finite sets $mathbbA$, of real numbers, with $f(mathbbA)subset mathbbA$.Prove that $int^1_0 f^-1 = 1 - int^1_0 f$Given a list of N integers, how to find out if the second derivative is positive or negative?Determine all real numbers $x in mathbbR$ for which the sequence $(a_n)_n in mathbbN$ with $a_0 = x $ converges and the associated limitsHow to show the sequence $a_n=ne^n,nge 1$ is increasing by comparison with a function who has domain on real numbers?Function that is continuous and monotone increasing but not differentiable at 0Definition of the tangent lineElegant way to make a bijection from the set of the complex numbers to the set of the real numbersWhat can we say about the function if it's derivative is Strictly increasing.$U subset mathbb R^n$ is an open set, $f in C^1(U, mathbb R^m)$. If $E subset U$ is a null set, then $f(E)$ is also a null set.Sets with a linear function to a proper subset
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find the not null, finite sets $mathbbA$, of real numbers, with $f(mathbbA)subset mathbbA$.
Prove that $int^1_0 f^-1 = 1 - int^1_0 f$Given a list of N integers, how to find out if the second derivative is positive or negative?Determine all real numbers $x in mathbbR$ for which the sequence $(a_n)_n in mathbbN$ with $a_0 = x $ converges and the associated limitsHow to show the sequence $a_n=ne^n,nge 1$ is increasing by comparison with a function who has domain on real numbers?Function that is continuous and monotone increasing but not differentiable at 0Definition of the tangent lineElegant way to make a bijection from the set of the complex numbers to the set of the real numbersWhat can we say about the function if it's derivative is Strictly increasing.$U subset mathbb R^n$ is an open set, $f in C^1(U, mathbb R^m)$. If $E subset U$ is a null set, then $f(E)$ is also a null set.Sets with a linear function to a proper subset
$begingroup$
I have $f:mathbbRrightarrow mathbbR$ with $f(x)=frac23(4^x-x^2-1)$. Show that f is increasing and find the not null, finite sets $A$, of real numbers, with $f(A)subset A$. I easily accomplished the first task by simply differentiating the function, but I don't understand what I am asked to do for the second and why it hepls me to know the monotonicity of the function.
calculus derivatives
$endgroup$
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$begingroup$
I have $f:mathbbRrightarrow mathbbR$ with $f(x)=frac23(4^x-x^2-1)$. Show that f is increasing and find the not null, finite sets $A$, of real numbers, with $f(A)subset A$. I easily accomplished the first task by simply differentiating the function, but I don't understand what I am asked to do for the second and why it hepls me to know the monotonicity of the function.
calculus derivatives
$endgroup$
add a comment |
$begingroup$
I have $f:mathbbRrightarrow mathbbR$ with $f(x)=frac23(4^x-x^2-1)$. Show that f is increasing and find the not null, finite sets $A$, of real numbers, with $f(A)subset A$. I easily accomplished the first task by simply differentiating the function, but I don't understand what I am asked to do for the second and why it hepls me to know the monotonicity of the function.
calculus derivatives
$endgroup$
I have $f:mathbbRrightarrow mathbbR$ with $f(x)=frac23(4^x-x^2-1)$. Show that f is increasing and find the not null, finite sets $A$, of real numbers, with $f(A)subset A$. I easily accomplished the first task by simply differentiating the function, but I don't understand what I am asked to do for the second and why it hepls me to know the monotonicity of the function.
calculus derivatives
calculus derivatives
asked Mar 15 at 6:19
user651754
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$begingroup$
Hint The second derivative of $f(x) - x$ is an increasing function, so $f$ has no more than three fixed points, but by inspection, $0, pm frac12$ are fixed points of $f$, so all of the (nonempty) subsets of $-frac12, 0, frac12$ meet the criterion. Now, show that for any point $a$ not in those subsets $a, f(a), f^2(a), ldots$ is a strictly increasing or strictly decreasing sequence, so that any set satisfying $A ni a$ and $f(A) subseteq A$ is infinite.
answered Mar 15 at 6:46
TravisTravis
63.7k769151
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$begingroup$
Hint The second derivative of $f(x) - x$ is an increasing function, so $f$ has no more than three fixed points, but by inspection, $0, pm frac12$ are fixed points of $f$, so all of the (nonempty) subsets of $-frac12, 0, frac12$ meet the criterion. Now, show that for any point $a$ not in those subsets $a, f(a), f^2(a), ldots$ is a strictly increasing or strictly decreasing sequence, so that any set satisfying $A ni a$ and $f(A) subseteq A$ is infinite.
answered Mar 15 at 6:46
TravisTravis
63.7k769151
$endgroup$
add a comment |
$begingroup$
Hint The second derivative of $f(x) - x$ is an increasing function, so $f$ has no more than three fixed points, but by inspection, $0, pm frac12$ are fixed points of $f$, so all of the (nonempty) subsets of $-frac12, 0, frac12$ meet the criterion. Now, show that for any point $a$ not in those subsets $a, f(a), f^2(a), ldots$ is a strictly increasing or strictly decreasing sequence, so that any set satisfying $A ni a$ and $f(A) subseteq A$ is infinite.
answered Mar 15 at 6:46
TravisTravis
63.7k769151
$endgroup$
add a comment |
$begingroup$
Hint The second derivative of $f(x) - x$ is an increasing function, so $f$ has no more than three fixed points, but by inspection, $0, pm frac12$ are fixed points of $f$, so all of the (nonempty) subsets of $-frac12, 0, frac12$ meet the criterion. Now, show that for any point $a$ not in those subsets $a, f(a), f^2(a), ldots$ is a strictly increasing or strictly decreasing sequence, so that any set satisfying $A ni a$ and $f(A) subseteq A$ is infinite.
answered Mar 15 at 6:46
TravisTravis
63.7k769151
$endgroup$
Hint The second derivative of $f(x) - x$ is an increasing function, so $f$ has no more than three fixed points, but by inspection, $0, pm frac12$ are fixed points of $f$, so all of the (nonempty) subsets of $-frac12, 0, frac12$ meet the criterion. Now, show that for any point $a$ not in those subsets $a, f(a), f^2(a), ldots$ is a strictly increasing or strictly decreasing sequence, so that any set satisfying $A ni a$ and $f(A) subseteq A$ is infinite.
answered Mar 15 at 6:46
TravisTravis
63.7k769151
answered Mar 15 at 6:46
TravisTravis
63.7k769151
answered Mar 15 at 6:46
TravisTravis
63.7k769151
answered Mar 15 at 6:46
TravisTravis
63.7k769151
63.7k769151
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