Câu hỏi

Cho hàm số y = f ( x ) có đạo hàm không âm trên [0;1], thỏa mãn f ( x ) > 0 với mọi x ∈ [ 0 ; 1 ] và . Nếu f ( 0 ) = 3 thì giá trị f ( 1 ) thuộc khoảng nào sau đây?

Cho hàm số $y = f (x)$ có đạo hàm không âm trên [0;1], thỏa mãn $f (x) > 0$ với mọi $x \in [0; 1]$ và . Nếu $f (0) = 3$ thì giá trị $f (1)$ thuộc khoảng nào sau đây?

R. Robo.Ctvx5

Giáo viên

Xác nhận câu trả lời

Giải thích

Ta có: +Nếu đặt +Nếu đặt

Ta có:

$left square bracket italic f left parenthesis italic x right parenthesis right square bracket squared ⋅ open square brackets italic f to the power of prime left parenthesis italic x right parenthesis close square brackets squared open parentheses italic x squared plus 1 close parentheses squared equals 1 plus left square bracket italic f left parenthesis italic x right parenthesis right square bracket squared left right double arrow fraction numerator left square bracket italic f left parenthesis italic x right parenthesis right square bracket squared ⋅ open square brackets italic f to the power of prime left parenthesis italic x right parenthesis close square brackets squared over denominator 1 plus left square bracket italic f left parenthesis italic x right parenthesis right square bracket squared end fraction equals 1 over open parentheses italic x squared plus 1 close parentheses squared$

$table attributes columnalign right columnspacing 0em end attributes row cell not stretchy left right double arrow fraction numerator italic f left parenthesis italic x right parenthesis ⋅ italic f to the power of prime left parenthesis italic x right parenthesis over denominator square root of 1 plus left square bracket italic f left parenthesis italic x right parenthesis right square bracket squared end root end fraction equals fraction numerator 1 over denominator italic x squared plus 1 end fraction end cell row cell not stretchy rightwards double arrow integral subscript 0 superscript 1 fraction numerator italic f left parenthesis italic x right parenthesis ⋅ italic f to the power of prime left parenthesis italic x right parenthesis over denominator square root of 1 plus left square bracket italic f left parenthesis italic x right parenthesis right square bracket squared end root end fraction italic d italic x equals integral subscript 0 superscript 1 fraction numerator 1 over denominator italic x squared plus 1 end fraction italic d italic x end cell row cell not stretchy rightwards double arrow integral subscript 0 superscript 1 fraction numerator italic f left parenthesis italic x right parenthesis ⋅ italic f to the power of prime left parenthesis italic x right parenthesis over denominator square root of 1 plus left square bracket italic f left parenthesis italic x right parenthesis right square bracket squared end root end fraction italic d italic x equals integral subscript 0 superscript 1 fraction numerator 1 over denominator italic x squared plus 1 end fraction italic d italic x end cell end table$

+Nếu đặt

$italic t equals square root of 1 plus left square bracket italic f left parenthesis italic x right parenthesis right square bracket squared end root not stretchy comes from italic d italic t equals fraction numerator italic f left parenthesis italic x right parenthesis ⋅ italic f to the power of prime left parenthesis italic x right parenthesis over denominator square root of 1 plus left square bracket italic f left parenthesis italic x right parenthesis right square bracket squared end root end fraction italic d italic x not stretchy comes from VT equals integral subscript italic 2 superscript square root of italic 1 italic plus italic f to the power of italic 2 italic left parenthesis italic 1 italic right parenthesis end root end superscript italic d italic t equals square root of italic 1 italic plus italic f to the power of italic 2 italic left parenthesis italic 1 italic right parenthesis end root − 2$

+Nếu đặt

$table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell text end text italic x equals tan invisible function application italic u not stretchy comes from italic d italic x equals open parentheses 1 plus tan squared invisible function application italic u close parentheses italic d italic u end cell row cell not stretchy comes from V P equals integral subscript 0 superscript italic pi over 4 end superscript fraction numerator 1 over denominator 1 plus tan squared invisible function application italic u end fraction open parentheses 1 plus tan squared invisible function application italic u close parentheses italic d italic x equals italic pi over 4 end cell row cell not stretchy rightwards double arrow square root of 1 plus italic f squared left parenthesis 1 right parenthesis end root − 2 equals italic pi over 4 end cell row cell not stretchy comes from italic f left parenthesis 1 right parenthesis equals square root of italic pi squared over 16 plus italic pi plus 3 end root almost equal to 2 comma 6 element of open parentheses 5 over 2 semicolon 3 close parentheses end cell end table$