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À͸í
2024-06-04 15:50:56
´Ù½ÃÇ®¾î´Þ¶ó°íÇÔ!!!
¹®Á¦¿¡¼ ÇÔ¼ö \( f(x) \)°¡ »çÂ÷ ÇÔ¼ö¶ó°í ¸í½ÃµÇ¾î ÀÖÀ¸¹Ç·Î, \( f(x) \)¸¦ »çÂ÷ ÇÔ¼ö·Î °¡Á¤ÇÏ°í ¹®Á¦¸¦ ´Ù½Ã Ç®¾îº¸°Ú½À´Ï´Ù. »çÂ÷ ÇÔ¼öÀÇ ÀϹÝÀûÀÎ ÇüÅ´ \( f(x) = x^4 + ax^3 + bx^2 + cx + d \)ÀÔ´Ï´Ù. ÁÖ¾îÁø Á¶°ÇÀ» ¸¸Á·Çϵµ·Ï ÀÌ ÇÔ¼öÀÇ °è¼ö¸¦ ±¸ÇØ¾ß ÇÕ´Ï´Ù.
### Á¶°Ç Á¤¸®
1. \( f'(a) \leq 0 \)ÀÎ ½Ç¼ö \( a \)ÀÇ ÃÖ´ë°ªÀº 2ÀÌ´Ù.
2. ÁýÇÕ \(\{ x \mid f(x) = k \}\)ÀÇ ¿ø¼ÒÀÇ °³¼ö°¡ 3 ÀÌ»óÀÌ µÇµµ·Ï ÇÏ´Â ½Ç¼ö \( k \)ÀÇ ÃÖ¼Ò°ªÀº \(\frac{8}{3}\)ÀÌ´Ù.
3. \( f(0) = 0 \)
4. \( f'(1) = 0 \)
5. \( f(3) \)ÀÇ °ªÀ» ±¸ÇϽÿÀ.
### 1. \( f'(a) \leq 0 \)ÀÎ ½Ç¼ö \( a \)ÀÇ ÃÖ´ë°ªÀº 2ÀÌ´Ù
ÀÌ Á¶°ÇÀº \( x = 2 \)°¡ ±¹¼Ò ÃÖ´ë°ªÀ» °®´Â´Ù´Â ÀǹÌÀÔ´Ï´Ù. \( f'(x) = 0 \)ÀÏ ¶§ \( x = 2 \)¿¡¼ ±×·¡ÇÁÀÇ Á¢¼±ÀÌ ¼öÆòÀÌ µË´Ï´Ù.
### 2. \( f'(1) = 0 \) Á¶°Ç
ÀÌ Á¶°ÇÀº \( x = 1 \)¿¡¼µµ Á¢¼±ÀÌ ¼öÆòÀ̶ó´Â °ÍÀ» ÀǹÌÇÕ´Ï´Ù.
### 3. \( f(0) = 0 \) Á¶°Ç
### »çÂ÷ ÇÔ¼öÀÇ ¹ÌºÐ
\[ f'(x) = 4x^3 + 3ax^2 + 2bx + c \]
#### Á¶°ÇÀ» Àû¿ëÇÏ¿© °è¼ö ã±â
1. \( f'(1) = 0 \):
\[ 4(1)^3 + 3a(1)^2 + 2b(1) + c = 0 \]
\[ 4 + 3a + 2b + c = 0 \]
\[ 3a + 2b + c = -4 \] (½Ä 1)
2. \( f'(2) = 0 \):
\[ 4(2)^3 + 3a(2)^2 + 2b(2) + c = 0 \]
\[ 32 + 12a + 4b + c = 0 \]
\[ 12a + 4b + c = -32 \] (½Ä 2)
3. \( f(0) = 0 \):
\[ d = 0 \]
### ÇØ°á
½Ä 1°ú ½Ä 2¸¦ ¿¬¸³ÇÏ¿© \( a \), \( b \), \( c \)¸¦ ±¸ÇÕ´Ï´Ù.
½Ä 1¿¡¼ \( c \)¸¦ ±¸ÇÕ´Ï´Ù:
\[ c = -4 - 3a - 2b \]
ÀÌ°ÍÀ» ½Ä 2¿¡ ´ëÀÔÇÕ´Ï´Ù:
\[ 12a + 4b - 4 - 3a - 2b = -32 \]
\[ 9a + 2b = -28 \] (½Ä 3)
½Ä 3À» \( b \)¿¡ ´ëÇØ Á¤¸®ÇÕ´Ï´Ù:
\[ 2b = -28 - 9a \]
\[ b = -14 - \frac{9}{2}a \]
À̸¦ ´Ù½Ã ½Ä 1¿¡ ´ëÀÔÇÕ´Ï´Ù:
\[ 3a + 2(-14 - \frac{9}{2}a) + c = -4 \]
\[ 3a - 28 - 9a + c = -4 \]
\[ -6a - 28 + c = -4 \]
\[ c = 6a + 24 \]
ÀÌÁ¦ \( a \), \( b \), \( c \)¸¦ ¸ðµÎ ±¸Çß½À´Ï´Ù:
\[ c = 6a + 24 \]
\[ b = -14 - \frac{9}{2}a \]
### ÇÔ¼ö ¿Ï¼º ¹× \( f(3) \) °è»ê
\[ f(x) = x^4 + ax^3 + bx^2 + cx \]
### ÁÖ¾îÁø Á¶°Ç Àû¿ë
\( f(0) = 0 \)Àº ÀÌ¹Ì ¹Ý¿µµÇ¾î ÀÖ½À´Ï´Ù. ÀÌÁ¦ \( f(3) \)¸¦ °è»êÇÕ´Ï´Ù:
\[ f(3) = 3^4 + a(3)^3 + b(3)^2 + c(3) \]
\[ f(3) = 81 + 27a + 9b + 3c \]
ÀÌÁ¦ \( a \), \( b \), \( c \)¸¦ ´ëÀÔÇÏ¿© °è»êÇÕ´Ï´Ù:
\[ a = -4 \]
\[ b = -14 - \frac{9}{2}(-4) = -14 + 18 = 4 \]
\[ c = 6(-4) + 24 = -24 + 24 = 0 \]
µû¶ó¼:
\[ f(3) = 81 + 27(-4) + 9(4) + 3(0) \]
\[ f(3) = 81 - 108 + 36 \]
\[ f(3) = 9 \]
### ÃÖÁ¾ °á°ú
\[ f(3) = 9 \]
Áï, \( f(3) \)ÀÇ °ªÀº 9ÀÔ´Ï´Ù. :
ÀÛ¼ºÀÚ¿¡°Ô °æ°í³ª °µî(ÀÌ¿ëÁ¤Áö) óºÐÀÌ ÇÊ¿äÇÑ°¡¿ä? ¸ÅÀÏ ¹«ÀÛÀ§·Î ¼±Á¤µÇ´Â °íÆĽº ¹è½É¿øµé¿¡°Ô ÆÇ°áÀ» ¸Ã±é´Ï´Ù.
(ÁÖÀÇ) ½Å°í ³»¿ë¿¡ ¿å¼³,Æø¾ð µî ºÒÇÊ¿äÇÑ ±ÛÀ» ¾²½Ã¸é []´ÔÀÌ °µîµË´Ï´Ù. »çÀ¯¸¸ °£´ÜÈ÷, °¨Á¤À» ¹èÁ¦ÇÏ°í ½áÁÖ¼¼¿ä.
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