calcul diferencial y integral: màxim y mínim

Si ( d_{x}[f(c)] = 0 & f(c)·d_{xx}^{2}[f(c)] < 0 ) ==> ...

... int[ 0 --> f(x) ]-[ x ] d[x] té un màxim en x = c

Si ( d_{x}[f(c)] = 0 & f(c)·d_{xx}^{2}[f(c)] > 0 ) ==> ...

... int[ 0 --> f(x) ]-[ x ] d[x] té un mínim en x = c

Si ( f(c) = 0 & d_{x}[f(c)]^{2} < 0 ) ==> ...

... int[ 0 --> f(x) ]-[ x ] d[x] té un màxim en x = c

Si ( f(c) = 0 & d_{x}[f(c)]^{2} > 0 ) ==> ...

... int[ 0 --> f(x) ]-[ x ] d[x] té un mínim en x = c

Demostració:

d_{x}[ int[ 0 --> f(x) ]-[ x ] d[x] ] = f(x)·d_{x}[f(x)]

d_{xx}^{2}[ int[ 0 --> f(x) ]-[ x ] d[x] ] = d_{x}[f(x)]^{2}+f(x)·d_{xx}^{2}[f(x)]

Exemples:

f(x) = i·( x+(-c) ) <==> d_{x}[f(x)] = i

int[ 0 --> f(x) ]-[ x ] d[x] = (-1)·(1/2)·( x+(-c) )^{2}

f(x) = (-i)·( x+(-c) ) <==> d_{x}[f(x)] = (-i)

int[ 0 --> f(x) ]-[ x ] d[x] = (-1)·(1/2)·( x+(-c) )^{2}

f(x) = ( x+(-c) ) <==> d_{x}[f(x)] = 1

int[ 0 --> f(x) ]-[ x ] d[x] = (1/2)·( x+(-c) )^{2}

f(x) = (-1)·( x+(-c) ) <==> d_{x}[f(x)] = (-1)

int[ 0 --> f(x) ]-[ x ] d[x] = (1/2)·( x+(-c) )^{2}

Si ( d_{x}[f(c)] = 0 & d_{xx}^{2}[f(c)] < 0 ) ==> ...

... int[ 0 --> f(x) ]-[ e^{x} ] d[x] té un màxim en x = c

Si ( d_{x}[f(c)] = 0 & d_{xx}^{2}[f(c)] > 0 ) ==> ...

... int[ 0 --> f(x) ]-[ e^{x} ] d[x] té un mínim en x = c

Demostració:

d_{x}[ int[ 0 --> f(x) ]-[ e^{x} ] d[x] ] = e^{f(x)}·d_{x}[f(x)]

d_{xx}^{2}[ int[ 0 --> f(x) ]-[ e^{x} ] d[x] ] = e^{f(x)}·d_{x}[f(x)]^{2}+e^{f(x)}·d_{xx}^{2}[f(x)]

Exemples:

d_{x}[f(x)] = ( c+(-x) ) <==> d_{xx}^{2}[f(x)] = (-1)

int[ 0 --> f(x) ]-[ e^{x} ] d[x] = e^{(-1)·(1/2)·( c+(-x) )^{2}}+(-1)

Inflexió: ( en x = c+1 || en x = c+(-1) )

d_{x}[f(x)] = ( x+(-c) ) <==> d_{xx}^{2}[f(x)] = 1

int[ 0 --> f(x) ]-[ e^{x} ] d[x] = e^{(1/2)·( x+(-c) )^{2}}+(-1)

Inflexió: ( en x = c+i || en x = c+(-i) )