Hướng dẫn giải Bài 1 (Trang 23, SGK Toán Giải tích 12)

Câu hỏi: Tìm GTLN, GTNN của hàm số: a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><msup><mi>x</mi><mn>3</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mn>9</mn><mi>x</mi><mo> </mo><mo>+</mo><mn>35</mn><mo> </mo></math>trên các đoạn <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mo>-</mo><mn>4</mn><mo>;</mo><mo> </mo><mn>4</mn></mrow></mfenced><mo> </mo><mi>v</mi><mi>à</mi><mo> </mo><mfenced open="[" close="]"><mrow><mn>0</mn><mo>;</mo><mo> </mo><mn>5</mn></mrow></mfenced></math> b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><msup><mi>x</mi><mn>4</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>+</mo><mn>2</mn></math> trên các đoạn <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mn>0</mn><mo> </mo><mo>;</mo><mn>3</mn></mrow></mfenced><mo> </mo><mi>v</mi><mi>à</mi><mo> </mo><mfenced open="[" close="]"><mrow><mn>2</mn><mo>;</mo><mo> </mo><mn>5</mn></mrow></mfenced></math> c) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mn>2</mn><mo> </mo><mo>-</mo><mi>x</mi></mrow><mrow><mn>1</mn><mo>-</mo><mi>x</mi><mo> </mo></mrow></mfrac></math> trên các đoạn <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mn>2</mn><mo> </mo><mo>;</mo><mo> </mo><mn>4</mn></mrow></mfenced><mo> </mo><mi>v</mi><mi>à</mi><mo> </mo><mfenced open="[" close="]"><mrow><mo>-</mo><mn>3</mn><mo> </mo><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn></mrow></mfenced></math> d) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><msqrt><mn>5</mn><mo> </mo><mo>-</mo><mo> </mo><mn>4</mn><mi>x</mi><mo> </mo></msqrt></math> trên đoạn <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mo>-</mo><mn>1</mn><mo> </mo><mo>;</mo><mo> </mo><mn>1</mn></mrow></mfenced></math> Hướng dẫn Giải: a) * Xét D = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mo>-</mo><mn>4</mn><mo> </mo><mo>;</mo><mo> </mo><mn>4</mn></mrow></mfenced></math>. HSLT trên <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mo>-</mo><mn>4</mn><mo> </mo><mo>;</mo><mo> </mo><mn>4</mn></mrow></mfenced></math>          <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mn>6</mn><mi>x</mi><mo> </mo><mo>-</mo><mo> </mo><mn>9</mn><mspace linebreak="newline"/><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo>[</mo><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mn>3</mn><mo> </mo><mo>∈</mo><mi>D</mi></mtd></mtr><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>1</mn><mo> </mo><mo>∈</mo><mi>D</mi></mtd></mtr></mtable></math> Ta có: y(-4) = -41; y(4) = 15; y(-1) = 40; y(3) = 8 Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mi>m</mi><mi>a</mi><mi>x</mi><mo> </mo><mi>y</mi><mo> </mo></mrow><mrow><mi>x</mi><mo> </mo><mo>∈</mo><mfenced open="[" close="]"><mrow><mo>-</mo><mn>4</mn><mo> </mo><mo>;</mo><mo> </mo><mn>4</mn></mrow></mfenced></mrow></munder><mo>=</mo><mo> </mo><mn>40</mn><mo>;</mo><mo> </mo><munder><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mo> </mo><mi>y</mi><mo> </mo></mrow><mrow><mi>x</mi><mo> </mo><mo>∈</mo><mfenced open="[" close="]"><mrow><mo>-</mo><mn>4</mn><mo> </mo><mo>;</mo><mo> </mo><mn>4</mn></mrow></mfenced></mrow></munder><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>41</mn></math>      * Xét D = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mn>0</mn><mo> </mo><mo>;</mo><mo> </mo><mn>5</mn></mrow></mfenced></math>. HSLT trên <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mn>0</mn><mo> </mo><mo>;</mo><mo> </mo><mn>5</mn></mrow></mfenced></math>       <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo>[</mo><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>3</mn><mo> </mo><mo>∈</mo><mi>D</mi></mtd></mtr><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>1</mn><mo> </mo><mo>∉</mo><mi>D</mi><mo> </mo></mtd></mtr></mtable></math>       Ta có y(0) = 35; y(5) = 40 ; y(3) =8       Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mi>m</mi><mi>a</mi><mi>x</mi><mo> </mo><mi>y</mi><mo> </mo></mrow><mrow><mi>x</mi><mo>∈</mo><mfenced open="[" close="]"><mrow><mn>0</mn><mo> </mo><mo>;</mo><mo> </mo><mn>5</mn></mrow></mfenced></mrow></munder><mo>=</mo><mn>40</mn><mo>;</mo><mo> </mo><munder><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mo> </mo><mi>y</mi></mrow><mrow><mi>x</mi><mo>∈</mo><mfenced open="[" close="]"><mrow><mn>0</mn><mo> </mo><mo>;</mo><mo> </mo><mn>5</mn></mrow></mfenced></mrow></munder><mo> </mo><mo>=</mo><mo> </mo><mn>8</mn></math> b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup><mo> </mo><mo>-</mo><mn>6</mn><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn><mi>x</mi><mfenced><mrow><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mn>3</mn></mrow></mfenced><mo>;</mo><mo> </mo><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo>[</mo><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></mtd></mtr><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><msqrt><mfrac><mn>3</mn><mn>2</mn></mfrac></msqrt></mtd></mtr><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><msqrt><mfrac><mn>3</mn><mn>2</mn></mfrac></msqrt></mtd></mtr></mtable></math> + Với D =<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mn>0</mn><mo> </mo><mo>;</mo><mo> </mo><mn>3</mn></mrow></mfenced></math>. thì x = <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><msqrt><mfrac><mn>3</mn><mn>2</mn></mfrac><mo> </mo></msqrt><mo> </mo><mo>∉</mo><mi>D</mi><mo> </mo></math>. HSLT trên <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mn>0</mn><mo> </mo><mo>;</mo><mo> </mo><mn>3</mn></mrow></mfenced></math>       Ta có y(0) = 2; y(3) = 56; y(<math xmlns="http://www.w3.org/1998/Math/MathML"><msqrt><mfrac><mn>3</mn><mn>2</mn></mfrac></msqrt></math>) = <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>          Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mo> </mo><mi>y</mi><mo> </mo></mrow><mrow><mfenced open="[" close="]"><mrow><mn>0</mn><mo> </mo><mo>;</mo><mo> </mo><mn>3</mn></mrow></mfenced><mo> </mo></mrow></munder><mo>=</mo><mo> </mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>;</mo><mo> </mo><munder><mrow><mi>m</mi><mi>a</mi><mi>x</mi><mo> </mo><mi>y</mi></mrow><mfenced open="[" close="]"><mrow><mn>0</mn><mo> </mo><mo>;</mo><mo> </mo><mn>3</mn></mrow></mfenced></munder><mo>=</mo><mn>56</mn></math> + Với D =<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mn>2</mn><mo> </mo><mo>;</mo><mo> </mo><mn>5</mn></mrow></mfenced></math>. HSLT trên <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mn>2</mn><mo> </mo><mo>;</mo><mo> </mo><mn>5</mn></mrow></mfenced></math> thì x = 0; x = <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>±</mo><msqrt><mfrac><mn>3</mn><mn>2</mn></mfrac></msqrt></math> đều không thuộc D nên y(2) = 6; y(5) = 552       Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mo> </mo><mi>y</mi><mo> </mo></mrow><mrow><mo>[</mo><mn>2</mn><mo> </mo><mo>;</mo><mo> </mo><mn>5</mn><mo>]</mo></mrow></munder><mo> </mo><mo>=</mo><mo> </mo><mn>6</mn><mo>;</mo><mo> </mo><mo> </mo><munder><mrow><mi>m</mi><mi>a</mi><mi>x</mi><mo> </mo><mi>y</mi><mo> </mo></mrow><mrow><mo>[</mo><mn>2</mn><mo>;</mo><mo> </mo><mn>5</mn><mo>]</mo></mrow></munder><mo>=</mo><mn>552</mn></math> c) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mi>x</mi><mo> </mo><mo>-</mo><mn>2</mn></mrow><mrow><mi>x</mi><mo> </mo><mo>-</mo><mn>1</mn></mrow></mfrac><mo>;</mo><mo> </mo><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac><mo> </mo><mo>></mo><mo> </mo><mn>0</mn><mo>;</mo><mo> </mo><mo>∀</mo><mi>x</mi><mo> </mo><mo>≠</mo><mo> </mo><mn>1</mn></math> + Với D = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mn>2</mn><mo> </mo><mo>;</mo><mo> </mo><mn>4</mn></mrow></mfenced></math>. HSLT trên<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mo>-</mo><mn>3</mn><mo> </mo><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn></mrow></mfenced></math>; y(2) =. ; y(4) = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>2</mn><mn>3</mn></mfrac></math>       Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mo> </mo><mi>y</mi><mo> </mo></mrow><mrow><mo>[</mo><mn>2</mn><mo> </mo><mo>;</mo><mo> </mo><mn>4</mn><mo>]</mo></mrow></munder><mo>=</mo><mn>0</mn><mo>;</mo><mo> </mo><munder><mrow><mi>m</mi><mi>a</mi><mi>x</mi><mo> </mo><mi>y</mi><mo> </mo></mrow><mrow><mo>[</mo><mn>2</mn><mo> </mo><mo>;</mo><mo> </mo><mn>4</mn><mo>]</mo></mrow></munder><mo>=</mo><mfrac><mn>2</mn><mn>3</mn></mfrac></math> + Với D = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mo>-</mo><mn>3</mn><mo> </mo><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn></mrow></mfenced></math>. HSLT trên <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mo>-</mo><mn>1</mn><mo> </mo><mo>;</mo><mo> </mo><mn>1</mn></mrow></mfenced></math>     y(-3) =<math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>5</mn><mn>4</mn></mfrac></math>; y(-2) = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>4</mn><mn>3</mn></mfrac></math>       Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mo> </mo><mi>y</mi><mo> </mo></mrow><mrow><mo>[</mo><mo>-</mo><mn>3</mn><mo> </mo><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>]</mo></mrow></munder><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>5</mn><mn>4</mn></mfrac><mo>;</mo><mo> </mo><mo> </mo><munder><mrow><mi>m</mi><mi>a</mi><mi>x</mi><mo> </mo><mi>y</mi><mo> </mo></mrow><mrow><mo>[</mo><mo>-</mo><mn>3</mn><mo> </mo><mo>;</mo><mo> </mo><mo>-</mo><mn>2</mn><mo>]</mo></mrow></munder><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>4</mn><mn>3</mn></mfrac></math> d) D =<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="[" close="]"><mrow><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>1</mn></mrow></mfenced></math>     <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mo>-</mo><mn>2</mn></mrow><msqrt><mn>5</mn><mo>-</mo><mn>4</mn><mi>x</mi><mo> </mo></msqrt></mfrac><mo><</mo><mn>0</mn><mo>,</mo><mo> </mo><mo>∀</mo><mi>x</mi><mo> </mo><mo>∈</mo><mo> </mo><mfenced open="[" close="]"><mrow><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>1</mn></mrow></mfenced></math>     y(-1) =. ; y(1) = 1    Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><munder><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mo> </mo><mi>y</mi><mo> </mo></mrow><mrow><mo>[</mo><mo>-</mo><mn>1</mn><mo>;</mo><mo> </mo><mn>1</mn><mo>]</mo></mrow></munder><mo>=</mo><mn>1</mn><mo> </mo><mo>;</mo><mo> </mo><munder><mrow><mi>m</mi><mi>a</mi><mi>x</mi><mo> </mo><mi>y</mi><mo> </mo></mrow><mrow><mo>[</mo><mo>-</mo><mn>1</mn><mo>;</mo><mn>1</mn><mo>]</mo></mrow></munder><mo> </mo><mo>=</mo><mo> </mo><mn>3</mn></math>