Piv txwv cov lus nug thiab kev sib tham txog kev nce cov haujlwm, txo cov haujlwm, thiab cov haujlwm ruaj khov
Cov kev ua lej yog ib qho kev kawm tob heev thiab muaj ntau yam ntxwv, ib qho yog yuav ua li cas lawv thiaj li raug soj ntsuam hauv kev nce, txo qis, lossis nyob ruaj khov. Kev paub seb ib qho kev ua lej puas nce, txo qis, lossis tas mus li ntawm ib qho sijhawm yog qhov tseem ceeb hauv ntau yam kev siv lej, suav nrog kev lag luam, physics, thiab engineering. Tsab xov xwm no yuav tham txog cov piv txwv thiab lawv cov kev sib tham ntsig txog kev nce, txo qis, thiab kev nyob ruaj khov.
Cov functions uas nce zuj zus, cov functions uas txo zuj zus, thiab cov functions uas nyob ruaj khov yog dab tsi?
1. Kev Ua Haujlwm Nce: Ib qho kev ua haujlwm \(f(x)\) tau hais tias yuav nce ntxiv rau ntawm ib qho sijhawm \(I\) yog tias rau txhua \(x_1\) thiab \(x_2\) hauv \(I\) nrog \(x_1 < x_2\), peb muaj \(f(x_1) \leq f(x_2)\). 2. Kev Ua Haujlwm Txo: Ntawm qhov tod tes, ib qho kev ua haujlwm \(f(x)\) tau hais tias yuav txo qis rau ntawm ib qho sijhawm \(I\) yog tias rau txhua \(x_1\) thiab \(x_2\) hauv \(I\) nrog \(x_1 < x_2\), peb muaj \(f(x_1) \geq f(x_2)\). 3. Kev Ua Haujlwm Ruaj Khoov: Ib qho kev ua haujlwm \(f(x)\) yog hais tias ruaj khoov ntawm ib qho ntu \(I\) yog tias rau txhua \(x\) hauv \(I\), qhov kev ua haujlwm muaj tib tus nqi, uas yog \(f(x) = c\) rau txhua \(x\) hauv \(I\), qhov twg c yog qhov tsis hloov pauv. Piv txwv li Lo Lus Nug 1: Txheeb Xyuas Lub Sijhawm Ntawm Ib Qho Kev Ua Haujlwm Nce Ntxiv Muab qhov kev ua haujlwm \(f(x) = 2x^3 - 3x^2 - 12x + 5\). Txheeb xyuas lub sijhawm uas qhov kev ua haujlwm nce ntxiv! Kev Sib Tham: Txhawm rau txiav txim siab lub sijhawm uas qhov kev ua haujlwm nce ntxiv, peb yuav tsum nrhiav thawj qhov derivative ntawm qhov kev ua haujlwm thiab tom qab ntawd tshuaj xyuas lub cim ntawm qhov derivative. 1. Kauj Ruam 1: Nrhiav thawj qhov derivative: \[ f'(x) = d/dx (2x^3 - 3x^2 - 12x + 5) \] \[ f'(x) = 6x^2 - 6x - 12 \] 2. Kauj Ruam 2: Txheeb xyuas cov ntsiab lus tseem ceeb: Cov ntsiab lus tseem ceeb yog cov ntsiab lus uas thawj qhov derivative yog xoom lossis tsis tau txhais. \[ 6x^2 - 6x - 12 = 0 \] Faib tag nrho cov kab zauv los ntawm 6: \[ x^2 - x - 2 = 0 \] Peb muab cov kab zauv quadratic no ua ke: \[ (x-2)(x+1) = 0 \] Yog li, cov ntsiab lus tseem ceeb yog \( x = 2 \) thiab \( x = -1 \). 3. Kauj Ruam 3: Txheeb xyuas lub cim ntawm thawj qhov derivative ntawm lub sijhawm tsim los ntawm cov ntsiab lus tseem ceeb: Peb yuav tsim ib lub rooj cim rau \(f'(x) \) ntawm lub sijhawm \((-\infty, -1) \), \((-1, 2) \), thiab \((2, \infty) \). - Rau \(x \in (-\infty, -1) \): Siv \(x = -2 \) \[ f'(-2) = 6(-2)^2 - 6(-2) - 12 = 24 + 12 - 12 = 24 \] Txij li thaum \(f'(-2) > 0 \), ces \(f(x) \) nce ntxiv rau lub sijhawm \((-\infty, -1) \).
- Rau \( x \in (-1, 2) \): Siv \( x = 0 \)
\[ f'(0) = 6(0)^2 – 6(0) – 12 = -12 \]
Vim tias \( f'(0) < 0 \), ces \( f(x) \) tab tom txo qis rau ntawm lub sijhawm \( (-1, 2) \). - Rau \( x \in (2, \infty) \): Siv \( x = 3 \) \[ f'(3) = 6(3)^2 - 6(3) - 12 = 54 - 18 - 12 = 24 \] Vim tias \( f'(3) > 0 \), ces \( f(x) \) tab tom nce rau ntawm lub sijhawm \( (2, \infty) \).
Yog li, qhov kev ua haujlwm \( f(x) \) nce ntxiv rau ntawm qhov sib nrug \( (-\infty, -1) \cup (2, \infty) \).
Piv Txwv Teeb Meem 2: Kev Txheeb Xyuas Qhov Kev Ua Haujlwm Txo Qis
Muab qhov kev ua haujlwm \( g(x) = 4x^4 – 8x^3 + 2 \). Txheeb xyuas cov sijhawm uas qhov kev ua haujlwm txo qis!
Kev Sib Tham:
1. Kauj Ruam 1: Nrhiav thawj qhov derivative:
\[ g'(x) = d/dx (4x^4 – 8x^3 + 2) \]
g'(x) = 16x^3 – 24x^2
2. Kauj Ruam 2: Txheeb xyuas qhov tseem ceeb tshaj plaws:
\[ 16x^3 – 24x^2 = 0 \]
\[ 8x^2(2x – 3) = 0 \]
Yog li ntawd, cov ntsiab lus tseem ceeb yog \( x = 0 \) thiab \( x = \frac{3}{2} \).
3. Kauj Ruam 3: Txheeb xyuas lub cim ntawm thawj qhov derivative ntawm qhov sib nrug:
- Rau \( x \in (-\infty, 0) \): Siv \( x = -1 \)
\[ g'(-1) = 16(-1)^3 – 24(-1)^2 = -16 – 24 = -40 \]
Vim tias \( g'(-1) < 0 \), ces \( g(x) \) tab tom txo qis rau ntawm lub sijhawm \( (-\infty, 0) \). - Rau \( x \in (0, \frac{3}{2}) \): Siv \( x = 1 \) \[ g'(1) = 16(1)^3 - 24(1)^2 = 16 - 24 = -8 \] Vim tias \( g'(1) < 0 \), ces \( g(x) \) tab tom txo qis rau ntawm lub sijhawm \( (0, \frac{3}{2}) \). - Rau \( x \in (\frac{3}{2}, \infty) \): Siv \( x = 2 \) \[ g'(2) = 16(2)^3 - 24(2)^2 = 128 - 96 = 32 \] Txij li thaum \( g'(2) > 0 \), ces \( g(x) \) nce ntxiv rau ntawm lub sijhawm \( (\frac{3}{2}, \infty) \).
Yog li, qhov kev ua haujlwm \(g(x) \) txo qis rau ntawm qhov sib txawv \((-\infty, 0) \cup (0, \frac{3}{2}) \).
Piv txwv lus nug 3: Txheeb xyuas lub sijhawm luv ntawm kev ua haujlwm thaum so
Muab qhov kev ua haujlwm \( h(x) = 7 \), txiav txim siab qhov sib nrug uas qhov kev ua haujlwm nyob ruaj khov!
Kev Sib Tham:
Ib qho kev ua haujlwm tas mus li xws li \( h(x) = 7 \) muaj thawj qhov derivative ntawm xoom rau txhua \( x \):
\[ h'(x) = 0 \]
Vim tias thawj qhov derivative yeej ib txwm yog xoom, qhov kev ua haujlwm yog nyob ruaj khov ntawm tag nrho thaj chaw, yog li peb tuaj yeem hais tias qhov kev ua haujlwm \(h(x) = 7\) yog nyob ruaj khov ntawm txhua tus lej tiag tiag, uas nyob rau hauv qhov cim qhia sijhawm yog \((-\infty,\infty)\).
Xaus
Kev nkag siab txog qhov sib txawv ntawm kev nce, txo, thiab cov xwm txheej ruaj khov ntawm ib qho kev ua haujlwm yog ib feem tseem ceeb ntawm kev tshuaj xyuas kev ua haujlwm. Los ntawm cov piv txwv saum toj no, peb tau tham txog cov ntsiab lus yooj yim thiab cov kauj ruam uas xav tau los nrhiav cov sib txawv no. Cov kev paub no muaj txiaj ntsig zoo heev rau ntau yam kev siv lej thiab kev xav.
