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==微分法(数学Ⅱ/教科書レベル基本問題5)==
• 最大値·最小値
• 関数のグラフと係数の符号 
• 実数解の個数(文字係数)
【例題1】
 次の関数について,( )内の範囲における最大値と最小値を求めてください.
 y=x3−3x+20≦x≦2
(解答)
y=3x2−3=3(x+1)(x−1)=0x=−1, 1
x···−1···
y+0
y4
0···1···2
0++
204
···
+

 増減表のうちで赤枠で示した範囲が定義域となる.
x=2のとき最大値4x=1のとき最小値0…(答)
• 「極大値」「極小値」の他に「区間の両端の値」を比べて,最大値,最小値とします.
• なお,x=0のときy=2となるが,この値はx=31.732...のときにもとり得る中間の値で,最大値でも最小値でもない.
• 「最大値,最小値を求めよ」という問題では,最大値,最小値の(yの)値だけでなく,その値を与えるxの値も答えるのが普通です.だから,答案は上記のように書くか,次の形に書くかのいずれかが普通.
 最大値4x=2),最小値0x=1
(参考)グラフは下図のようになります
【例題2】
 次の関数について,( )内の範囲における最大値と最小値を求めてください.
 y=−x3−3x2+4−2≦x≦1
(解答)
y=−3x2−6x=−3x(x+2)=0x=−2, 0
x···
y
y
−2···0···1
0+0
040
···

 増減表のうちで赤枠で示した範囲が定義域となる.
x=0のとき最大値4x=−2, 1のとき最小値0…(答)
• 同じ値が最小値となっているとき,「どちらも最小値」とします.「日本タイ記録」「世界タイ記録」のような考え方です
(参考)グラフは下図のようになります

• 空欄を「半角数字(1, 2, 3 など)」「半角英小文字」(a, b, c など)で埋めて,採点ボタンを押してください.
• 採点すれば採点結果と解説が出ます.見ているだけでは解説は出ません.
次の関数について,( )内の範囲における最大値と最小値を求めてください.
【問題1-1】
 y=2x3−3x2−12x−2≦x≦3
x=のとき最大値
x=のとき最小値
採点する

次の関数について,( )内の範囲における最大値と最小値を求めてください.
【問題1-2】
 y=−x3+x2−1≦x≦1
x=のとき最大値
x=のとき最小値
採点する

《関数のグラフと係数の符号》
【例題3】
 3次関数f(x)=ax3+bx2+cx+dのグラフが下図のような形になるとき,係数a, b, c, dの符号を定めてください.
α
β
(解答)
f(x)=ax3+bx2+cx+d…(1)
f(x)=3ax2+2bx+c…(2)
y軸との交点を見ると,f(0)=d>0
x→−∞のときf(x)→−∞x→∞のときf(x)→∞だからa>0
• グラフから,極大値と極小値があるから,f(x)=3ax2+2bx+c=0の2つの実数解を0<α<βとおくと,(2)について解と係数の関係から
α+β=2b3a>0
αβ=c3a>0
a>0だから,b<0, c>0
以上から,a>0, b<0, c>0, d>0…(答)

【問題3-1】
 3次関数f(x)=ax3+bx2+cx+dのグラフが下図のような形になるとき,係数a, b, c, dの符号を定めてください.
(各々正しい方をクリックしてください)
α
β
a>0b>0c>0d>0
a<0b<0c<0d<0

解説を読む

【問題3-2】
 3次関数f(x)=ax3+bx2+cx+dのグラフが下図のような形になるとき,係数a, b, c, dの符号を定めてください.
(各々正しい方をクリックしてください)
α
β
a>0b>0c>0d>0
a<0b<0c<0d<0

解説を読む

【問題3-3】
 3次関数f(x)=ax3+bx2+cx+dのグラフが下図のような形になるとき,係数a, b, c, dの符号を定めてください.
(各々正しい方をクリックしてください)
α
β
a>0b>0c>0d>0
a<0b<0c<0d<0

解説を読む

《文字係数方程式の実数解の個数》
(1) 文字定数aの値に応じて,方程式x3−3x+a=0の実数解の個数を調べるような問題では
a=−x3+3xと変形することにより,2つのグラフf(x)=ag(x)=−x3+3xの共有点の個数を調べるとよい
(2) x3−3ax2+4a=0a>0)のように,文字定数aについて解いてしまうと
a=x33x24
といった「分数関数」になってしまう場合は,数学Ⅱの範囲では導関数を求められないので,元の関数のまま
   y=x3−3ax2+4a
の形を微分して,文字定数を含めた形でx軸との共有点の個数を調べるとよい.
※(2)の方が適用範囲が広いが,途中の処理がやや複雑になる.
(1)の場合,黒線で示したg(x)=−x3+3xのグラフを描いてから,文字定数aの値の大小に応じて,赤線で示したf(x)=aのグラフを描くと,共有点の個数が調べられる.
a<−2
−2<a<2
a>2
a=−2
a=2

【例題4】
 aを正の定数とするとき,3次方程式x3−3ax2+4a=0の実数解の個数を調べてください.
(解答)
y=x3−3ax2+4aとおく
y=3x2−6ax=3x(x−2a)
y=0x=0, 2a
a>0だから,増減表は次の通り
x···0···2a···
y+00+
y4a−4a3+4a
ここで,a>0のとき
ア) −4a3+4a=−4a(a+1)(a−1)>0a>1(下図赤)
イ) −4a3+4a=−4a(a+1)(a−1)=0a=1(下図緑)
ウ) −4a3+4a=−4a(a+1)(a−1)<00<a<1(下図青)
x軸との共有点の個数は
ア) a>1のとき,1個
イ) a=1のとき,2個
ウ) 0<a<1のとき,3個…(答)

【問題4-1】
 3次方程式2x3−3ax2+1=0が異なる3つの実数解をもつとき,定数aの値の範囲を求めてください.
 a>
採点する

【問題4-2】
 3次方程式2x3−6x+a=0が負の解を2つと正の解を1つもつとき,定数aの値の範囲を求めてください.
 <a<
採点する

【問題4-3】
 3次方程式x33x2x+1=2x+aが負の解を1つと正の解を2つもつとき,定数aの値の範囲を求めてください.
 <a<
採点する

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