Mixing
There is c such that $f''(c)g(c)+f(c)g''(c)+f'(c)g'(c)=0$
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Let $f,g:[a,b]tomathbb{R}$ both increasing, twice differentiable and $g$ is also convex, $f(a)=0$, $g(b)=0$, $f^prime(a)=0$. Prove that there exists $cin(a,b)$ such that
$$f''(c)g(c)+f(c)g''(c)+f'(c)g'(c)=0$$
I found that if we denote by $h(x)=f''(x)g(x)+f(x)g''(x)+f'(x)g'(x)$, we have that $h$ has the intermediate value property and, also, $h(a)leq 0$ and $h(b)>0$. I'm stuck only if $h(a)=0$.
real-analysis derivatives definite-integrals
asked Jan 17 at 20:28
Andrew VAndrew V
30911
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add a comment |
$begingroup$
Let $f,g:[a,b]tomathbb{R}$ both increasing, twice differentiable and $g$ is also convex, $f(a)=0$, $g(b)=0$, $f^prime(a)=0$. Prove that there exists $cin(a,b)$ such that
$$f''(c)g(c)+f(c)g''(c)+f'(c)g'(c)=0$$
I found that if we denote by $h(x)=f''(x)g(x)+f(x)g''(x)+f'(x)g'(x)$, we have that $h$ has the intermediate value property and, also, $h(a)leq 0$ and $h(b)>0$. I'm stuck only if $h(a)=0$.
real-analysis derivatives definite-integrals
asked Jan 17 at 20:28
Andrew VAndrew V
30911
$endgroup$
$begingroup$
Are you sure you do not have a coefficient "2" in front of $f'(c)g'(c)$?
$endgroup$
– GReyes
Jan 17 at 22:40
$begingroup$
I'm afraid I am
$endgroup$
– Andrew V
Jan 17 at 23:25
add a comment |
$begingroup$
Let $f,g:[a,b]tomathbb{R}$ both increasing, twice differentiable and $g$ is also convex, $f(a)=0$, $g(b)=0$, $f^prime(a)=0$. Prove that there exists $cin(a,b)$ such that
$$f''(c)g(c)+f(c)g''(c)+f'(c)g'(c)=0$$
I found that if we denote by $h(x)=f''(x)g(x)+f(x)g''(x)+f'(x)g'(x)$, we have that $h$ has the intermediate value property and, also, $h(a)leq 0$ and $h(b)>0$. I'm stuck only if $h(a)=0$.
real-analysis derivatives definite-integrals
asked Jan 17 at 20:28
Andrew VAndrew V
30911
$endgroup$
Let $f,g:[a,b]tomathbb{R}$ both increasing, twice differentiable and $g$ is also convex, $f(a)=0$, $g(b)=0$, $f^prime(a)=0$. Prove that there exists $cin(a,b)$ such that
$$f''(c)g(c)+f(c)g''(c)+f'(c)g'(c)=0$$
I found that if we denote by $h(x)=f''(x)g(x)+f(x)g''(x)+f'(x)g'(x)$, we have that $h$ has the intermediate value property and, also, $h(a)leq 0$ and $h(b)>0$. I'm stuck only if $h(a)=0$.
real-analysis derivatives definite-integrals
real-analysis derivatives definite-integrals
asked Jan 17 at 20:28
Andrew VAndrew V
30911
asked Jan 17 at 20:28
Andrew VAndrew V
30911
asked Jan 17 at 20:28
Andrew VAndrew V
30911
asked Jan 17 at 20:28
Andrew VAndrew V
30911
asked Jan 17 at 20:28
Andrew VAndrew V
30911
30911
$begingroup$
Are you sure you do not have a coefficient "2" in front of $f'(c)g'(c)$?
$endgroup$
– GReyes
Jan 17 at 22:40
$begingroup$
I'm afraid I am
$endgroup$
– Andrew V
Jan 17 at 23:25
add a comment |
$begingroup$
Are you sure you do not have a coefficient "2" in front of $f'(c)g'(c)$?
$endgroup$
– GReyes
Jan 17 at 22:40
$begingroup$
I'm afraid I am
$endgroup$
– Andrew V
Jan 17 at 23:25
$begingroup$
Are you sure you do not have a coefficient "2" in front of $f'(c)g'(c)$?
$endgroup$
– GReyes
Jan 17 at 22:40
$begingroup$
Are you sure you do not have a coefficient "2" in front of $f'(c)g'(c)$?
$endgroup$
– GReyes
Jan 17 at 22:40
$begingroup$
I'm afraid I am
$endgroup$
– Andrew V
Jan 17 at 23:25
$begingroup$
I'm afraid I am
$endgroup$
– Andrew V
Jan 17 at 23:25
add a comment |
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$begingroup$
Are you sure you do not have a coefficient "2" in front of $f'(c)g'(c)$?
$endgroup$
– GReyes
Jan 17 at 22:40
$begingroup$
I'm afraid I am
$endgroup$
– Andrew V
Jan 17 at 23:25