Matematica - Functii continue. Functii derivabile

Functii continue. Functii derivabile - Intrebarea 13

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Se consideră funcția $f colon open parentheses negative infinity comma 2 close parentheses rightwards arrow straight real numbers comma space f left parenthesis x right parenthesis equals fraction numerator e to the power of negative x end exponent over denominator x minus 2 end fraction$ . Să se precizeze care dintre următoarele afirmații este adevărată.

f left parenthesis x right parenthesis less or equal than 1 over e

f left parenthesis x right parenthesis greater or equal than negative 1 over e

f left parenthesis x right parenthesis less or equal than negative 1 over e

Raspunsul este corect.

Raspunsul este gresit.

Explicatie:

Răspuns corect: C

$f apostrophe left parenthesis x right parenthesis equals 0 rightwards double arrow x equals 1$

$f open parentheses 1 close parentheses equals negative 1 over e stack l i m with x rightwards arrow negative infinity below f left parenthesis x right parenthesis equals negative infinity stack l i m with table row cell x rightwards arrow 2 end cell row cell x less than 2 end cell end table below f left parenthesis x right parenthesis equals negative infinity$

Realizăm tabelul de monotonie și avem:

$f apostrophe left parenthesis x right parenthesis greater or equal than 0 comma space open parentheses for all close parentheses x element of left parenthesis negative infinity semicolon 1 right square bracket rightwards double arrow$ $f$ este crescătoare pe $left parenthesis negative infinity semicolon 1 right square bracket$

$f apostrophe left parenthesis x right parenthesis less or equal than 0 comma space open parentheses for all close parentheses x element of left square bracket 1 semicolon 2 right parenthesis rightwards double arrow$ $f$ este descrescătoare pe $left square bracket 1 semicolon 2 right parenthesis$

$rightwards double arrow x equals 1$ este punct de maxim global

$rightwards double arrow f left parenthesis x right parenthesis less or equal than f left parenthesis 1 right parenthesis comma space open parentheses for all close parentheses x element of left parenthesis negative infinity semicolon 2 right parenthesis$

$left right double arrow f left parenthesis x right parenthesis less or equal than negative 1 over e comma space open parentheses for all close parentheses x element of left parenthesis negative infinity semicolon 2 right parenthesis$