Study of an implicit function
$begingroup$
The problem consists of two questions :
The first asks to prove that for any real number $x$, there exists a unique real numbers $t$ such that
$$t_x^3x^2+t_x+x=0$$
I'm having problems with the second question which asks to study the implicit function $f:xlongmapsto t_x$
A hint asked to study the inverse function of $f$ but I'm having a hard time to prove that $f$ is invertible, let alone to study its inverse.
What I could do :
$f(x)$ and $x$ have opposite signs due to the equation $t_x(x^2t_x^2+1)=-x$
- seeing the equation as a quadratic in $x$, the discriminant should be $ge 0$ meaning range of $f$ is within $left[-frac{1}{sqrt 2}, frac{1}{sqrt 2} right]$
- if $f$ is invertible the inverse has one of the expressions
$dfrac{-1pmsqrt{1-4t^4}}{2t^3}$ (got by solving $(g(y))^2y^3+g(y)+y=0$) - A few tests on wolframalpha show that $f$ would be odd, negative and decreasing on $[0,+infty)$.
Any help or recommendations to treat such problems would be appreciated.
calculus implicit-function
asked Jan 2 at 9:16
Oussama SarihOussama Sarih
49027
$endgroup$
add a comment |
$begingroup$
The problem consists of two questions :
The first asks to prove that for any real number $x$, there exists a unique real numbers $t$ such that
$$t_x^3x^2+t_x+x=0$$
I'm having problems with the second question which asks to study the implicit function $f:xlongmapsto t_x$
A hint asked to study the inverse function of $f$ but I'm having a hard time to prove that $f$ is invertible, let alone to study its inverse.
What I could do :
$f(x)$ and $x$ have opposite signs due to the equation $t_x(x^2t_x^2+1)=-x$
- seeing the equation as a quadratic in $x$, the discriminant should be $ge 0$ meaning range of $f$ is within $left[-frac{1}{sqrt 2}, frac{1}{sqrt 2} right]$
- if $f$ is invertible the inverse has one of the expressions
$dfrac{-1pmsqrt{1-4t^4}}{2t^3}$ (got by solving $(g(y))^2y^3+g(y)+y=0$) - A few tests on wolframalpha show that $f$ would be odd, negative and decreasing on $[0,+infty)$.
Any help or recommendations to treat such problems would be appreciated.
calculus implicit-function
asked Jan 2 at 9:16
Oussama SarihOussama Sarih
49027
$endgroup$
$begingroup$
But the fact that for any $x$ there exists a unique $t$ doesn't mean that $x mapsto t_x $ is injective. Regarding the question; the original problem is that "vague". It asks the "study" of the function, that might include limits, continuity, monotonicity, and maybe other more features of $f$.
$endgroup$
– Oussama Sarih
Jan 2 at 10:22
add a comment |
$begingroup$
The problem consists of two questions :
The first asks to prove that for any real number $x$, there exists a unique real numbers $t$ such that
$$t_x^3x^2+t_x+x=0$$
I'm having problems with the second question which asks to study the implicit function $f:xlongmapsto t_x$
A hint asked to study the inverse function of $f$ but I'm having a hard time to prove that $f$ is invertible, let alone to study its inverse.
What I could do :
$f(x)$ and $x$ have opposite signs due to the equation $t_x(x^2t_x^2+1)=-x$
- seeing the equation as a quadratic in $x$, the discriminant should be $ge 0$ meaning range of $f$ is within $left[-frac{1}{sqrt 2}, frac{1}{sqrt 2} right]$
- if $f$ is invertible the inverse has one of the expressions
$dfrac{-1pmsqrt{1-4t^4}}{2t^3}$ (got by solving $(g(y))^2y^3+g(y)+y=0$) - A few tests on wolframalpha show that $f$ would be odd, negative and decreasing on $[0,+infty)$.
Any help or recommendations to treat such problems would be appreciated.
calculus implicit-function
asked Jan 2 at 9:16
Oussama SarihOussama Sarih
49027
$endgroup$
The problem consists of two questions :
The first asks to prove that for any real number $x$, there exists a unique real numbers $t$ such that
$$t_x^3x^2+t_x+x=0$$
I'm having problems with the second question which asks to study the implicit function $f:xlongmapsto t_x$
A hint asked to study the inverse function of $f$ but I'm having a hard time to prove that $f$ is invertible, let alone to study its inverse.
What I could do :
$f(x)$ and $x$ have opposite signs due to the equation $t_x(x^2t_x^2+1)=-x$
- seeing the equation as a quadratic in $x$, the discriminant should be $ge 0$ meaning range of $f$ is within $left[-frac{1}{sqrt 2}, frac{1}{sqrt 2} right]$
- if $f$ is invertible the inverse has one of the expressions
$dfrac{-1pmsqrt{1-4t^4}}{2t^3}$ (got by solving $(g(y))^2y^3+g(y)+y=0$) - A few tests on wolframalpha show that $f$ would be odd, negative and decreasing on $[0,+infty)$.
Any help or recommendations to treat such problems would be appreciated.
calculus implicit-function
calculus implicit-function
asked Jan 2 at 9:16
Oussama SarihOussama Sarih
49027
asked Jan 2 at 9:16
Oussama SarihOussama Sarih
49027
asked Jan 2 at 9:16
Oussama SarihOussama Sarih
49027
asked Jan 2 at 9:16
Oussama SarihOussama Sarih
49027
asked Jan 2 at 9:16
Oussama SarihOussama Sarih
49027
49027
$begingroup$
But the fact that for any $x$ there exists a unique $t$ doesn't mean that $x mapsto t_x $ is injective. Regarding the question; the original problem is that "vague". It asks the "study" of the function, that might include limits, continuity, monotonicity, and maybe other more features of $f$.
$endgroup$
– Oussama Sarih
Jan 2 at 10:22
add a comment |
$begingroup$
But the fact that for any $x$ there exists a unique $t$ doesn't mean that $x mapsto t_x $ is injective. Regarding the question; the original problem is that "vague". It asks the "study" of the function, that might include limits, continuity, monotonicity, and maybe other more features of $f$.
$endgroup$
– Oussama Sarih
Jan 2 at 10:22
$begingroup$
But the fact that for any $x$ there exists a unique $t$ doesn't mean that $x mapsto t_x $ is injective. Regarding the question; the original problem is that "vague". It asks the "study" of the function, that might include limits, continuity, monotonicity, and maybe other more features of $f$.
$endgroup$
– Oussama Sarih
Jan 2 at 10:22
$begingroup$
But the fact that for any $x$ there exists a unique $t$ doesn't mean that $x mapsto t_x $ is injective. Regarding the question; the original problem is that "vague". It asks the "study" of the function, that might include limits, continuity, monotonicity, and maybe other more features of $f$.
$endgroup$
– Oussama Sarih
Jan 2 at 10:22
add a comment |
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$begingroup$
But the fact that for any $x$ there exists a unique $t$ doesn't mean that $x mapsto t_x $ is injective. Regarding the question; the original problem is that "vague". It asks the "study" of the function, that might include limits, continuity, monotonicity, and maybe other more features of $f$.
$endgroup$
– Oussama Sarih
Jan 2 at 10:22