Hướng dẫn giải Bài 1 (Trang 18 SGK Toán Giải tích 12)

Câu hỏi: Áp dụng Quy tắc I, hãy tìm các điểm cực trị của hàm số sau: a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mo> </mo><mo>+</mo><mn>3</mn><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>36</mn><mi>x</mi><mo> </mo><mo>-</mo><mn>10</mn></math> b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><msup><mi>x</mi><mn>4</mn></msup><mo> </mo><mo>+</mo><mn>2</mn><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mn>3</mn></math> c) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mfrac><mn>1</mn><mi>x</mi></mfrac></math> d) <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><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup></math> e) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mi>x</mi><mo> </mo><mo>+</mo><mn>1</mn></msqrt></math>   Hướng dẫn Giải: a) TXD: D = R <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mn>6</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>36</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><mo> </mo><mn>2</mn><mo> </mo><mfenced><mrow><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>54</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>3</mn><mo> </mo><mfenced><mrow><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mn>71</mn></mrow></mfenced></mtd></mtr></mtable></math> Bảng biến thiên: <img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/18022022/anh-chup-man-hinh-2022-02-18-luc-131100-JDTyFa.png" width="545" height="99" /> Hàm số đạt cực đại tại điểm x = -3 và  <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mi>C</mi><mi>D</mi></mrow></msub><mo> </mo><mo>=</mo><mo> </mo><mn>71</mn></math> Hàm số đạt cực tiểu tại điểm x = 2 và <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mi>C</mi><mi>T</mi></mrow></msub><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>54</mn></math> b) TXD: D = R <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mn>4</mn><msup><mi>x</mi><mn>3</mn></msup><mo> </mo><mo>+</mo><mn>4</mn><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>4</mn><mi>x</mi><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mspace linebreak="newline"/><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇔</mo><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mfenced><mrow><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>3</mn></mrow></mfenced></math> Bảng biến thiên: <img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/18022022/anh-chup-man-hinh-2022-02-18-luc-131356-amUr11.png" width="567" height="103" /> Hàm số có điểm cực tiểu tại x = 0 và <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mi>C</mi><mi>T</mi></mrow></msub><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>3</mn></math> c) TXD: D = R \ {0} <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo> </mo><mo>-</mo><mo> </mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>2</mn></msup></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn></mrow><msup><mi>x</mi><mn>2</mn></msup></mfrac><mspace linebreak="newline"/><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mn>1</mn><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo> </mo><mo>[</mo><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>1</mn><mo> </mo><mfenced><mrow><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>1</mn><mo> </mo><mfenced><mrow><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mn>2</mn></mrow></mfenced></mtd></mtr></mtable></math> Bảng biến thiên: <img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/18022022/anh-chup-man-hinh-2022-02-18-luc-131636-kKkQI1.png" width="604" height="110" /> Hàm số đạt cực đại tại x = -1, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mi>C</mi><mi>Đ</mi><mo> </mo></mrow></msub><mo>=</mo><mo> </mo><mo>-</mo><mn>2</mn></math> Hàm số đạt cực tiểu tại x = 1, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mi>C</mi><mi>T</mi></mrow></msub><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn></math> d) TXD: D = R  <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><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mn>2</mn></msup><mo> </mo><mo>-</mo><mn>2</mn><msup><mi>x</mi><mn>3</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><msup><mi>x</mi><mn>2</mn></msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mfenced><mrow><mn>3</mn><mo>-</mo><mn>5</mn><mi>x</mi></mrow></mfenced><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>1</mn><mo> </mo><mfenced><mrow><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mfrac><mn>3</mn><mn>5</mn></mfrac><mo> </mo><mfenced><mrow><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>108</mn><mn>3125</mn></mfrac></mrow></mfenced></mtd></mtr><mtr><mtd><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></mtd></mtr></mtable></math> Bảng biến thiên: <img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/18022022/anh-chup-man-hinh-2022-02-18-luc-132020-9EhQmQ.png" width="569" height="122" /> Hàm số đạt cực đại tại điểm x = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>3</mn><mn>5</mn></mfrac><mo>,</mo><mo> </mo><msub><mi>y</mi><mrow><mi>C</mi><mi>Đ</mi></mrow></msub><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>108</mn><mn>3125</mn></mfrac></math> Hàm số đạt cực tiểu tại điểm x = 1, <math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>y</mi><mrow><mi>C</mi><mi>T</mi></mrow></msub><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></math> e) Vì <math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn><mo>></mo><mn>0</mn><mo>,</mo><mo>∀</mo><mi>x</mi><mo> </mo><mo>∈</mo><mo> </mo><mi>R</mi><mo> </mo></math> nên tập TXD: D = R <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mn>2</mn><mi>x</mi><mo> </mo><mo>-</mo><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><msup><mi>x</mi><mn>2</mn></msup><mo> </mo><mo>-</mo><mo> </mo><mi>x</mi><mo> </mo><mo>+</mo><mo> </mo><mn>1</mn></msqrt></mrow></mfrac><mspace linebreak="newline"/><mi>y</mi><mo>'</mo><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn><mo> </mo><mo>⇔</mo><mo> </mo><mi>x</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo> </mo><mfenced><mrow><mi>y</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></mrow></mfenced></math> Bảng biến thiên: <img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/18022022/anh-chup-man-hinh-2022-02-18-luc-132335-LqgVwS.png" width="574" height="123" /> Hàm số đạt cực tiểu tại điểm x = <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo> </mo><msub><mi>y</mi><mrow><mi>C</mi><mi>T</mi></mrow></msub><mo> </mo><mo>=</mo><mo> </mo><mfrac><msqrt><mn>3</mn></msqrt><mn>2</mn></mfrac></math>