1.次のグラフを描きなさい。
(1)\(f(x)=xe^x\)
\(f'(x)=e^x+xe^x=(1+x)e^x\)
\(f''(x)=e^x+(1+x)e^x=(2+x)e^x\)
増減表にまとめると、
| \(x\) | \(\cdots\) | \(-2\) | \(\cdots\) | \(-1\) | \(\cdots\) |
| \(f'(x)\) | \(-\) | \(-\) | \(-\) | \(0\) | \(+\) |
| \(f''(x)\) | \(-\) | \(0\) | \(+\) | \(+\) | \(+\) |
| \(f(x)\) | \(⤵\) | \(\displaystyle -\frac{2}{e^2}\) | \(⤷\) | \(\displaystyle -\frac{1}{e}\) | \(⤴\) |
\(\displaystyle \lim_{x\to-\infty}y=0\)より、
\(x\)軸は漸近線である。
(2)\(f(x)=e^{-2x^2}\)
\(f'(x)=-4xe^{-2x^2}\)
\(f''(x)=-4e^{-2x^2}+16x^2e^{-2x^2}\)
\(=-4e^{-2x^2}(1-4x^2)\)
増減表にまとめると、
| \(x\) | \(\cdots\) | \(\displaystyle -\frac{1}{2}\) | \(\cdots\) | \(0\) | \(\cdots\) | \(\displaystyle \frac{1}{2}\) | \(\cdots\) |
| \(f'(x)\) | \(+\) | \(+\) | \(+\) | \(0\) | \(-\) | \(-\) | \(-\) |
| \(f''(x)\) | \(+\) | \(0\) | \(-\) | \(-\) | \(-\) | \(0\) | \(+\) |
| \(f(x)\) | \(⤴\) | \(\displaystyle \frac{1}{\sqrt{e}}\) | \(↱\) | \(1\) | \(⤵\) | \(\displaystyle \frac{1}{\sqrt{e}}\) | \(⤷\) |
\(\displaystyle \lim_{x\to-\infty}y=0,\lim_{x\to\infty}y=0\)より、
\(x\)軸は漸近線である。
(3)\(\displaystyle f(x)=\frac{x^2-x+2}{x+1}\)
\(\displaystyle f'(x)=\frac{(2x-1)(x+1)-(x^2-x+2)}{(x+1)^2}\)
\(\displaystyle =\frac{(x+3)(x-1)}{(x+1)^2}\)
\(\displaystyle f''(x)=\frac{(2x+2)(x+1)^2-(x^2+2x-3)}{(x+1)^4}\)
\(\displaystyle =\frac{8}{(x+1)^3}\)
増減表にまとめると、
| \(x\) | \(\cdots\) | \(-3\) | \(\cdots\) | \(-1\) | \(\cdots\) | \(1\) | \(\cdots\) |
| \(f'(x)\) | \(+\) | \(0\) | \(-\) | \(-\) | \(0\) | \(+\) | |
| \(f''(x)\) | \(-\) | \(-\) | \(-\) | \(+\) | \(+\) | \(+\) | |
| \(f(x)\) | \(↱\) | \(-7\) | \(⤵\) | \(⤷\) | \(1\) | \(⤴\) |
\(\displaystyle \lim_{x\to-1+0}y=\infty,\lim_{x\to-1-0}y=-\infty\)より、
\(x=-1\)は漸近線である。
\(\displaystyle f(x)=\frac{4}{x+1}+x-2\)より、
\(y=x-2\)は漸近線である。
2.次の関数の極値を求めなさい。
(1)\(f(x)=x^4+2x^3+1\)
\(f'(x)=4x^3+6x^2=2x^2(2x+3)\)
\(f''(x)=12x^2+12x=12x(x+1)\)
増減表にまとめると、
| \(x\) | \(\cdots\) | \(\displaystyle -\frac{3}{2}\) | \(\cdots\) | \(-1\) | \(\cdots\) | \(0\) | \(\cdots\) |
| \(f'(x)\) | \(-\) | \(0\) | \(+\) | \(+\) | \(+\) | \(+\) | \(+\) |
| \(f''(x)\) | \(+\) | \(+\) | \(+\) | \(0\) | \(-\) | \(0\) | \(+\) |
| \(f(x)\) | \(⤷\) | \(\displaystyle -\frac{11}{16}\) | \(⤴\) | \(0\) | \(↱\) | \(1\) | \(⤴\) |
したがって、
\(\displaystyle x=-\frac{3}{2}\)のとき、極小値\(\displaystyle -\frac{11}{16}\)
(2)\(f(x)=x-\cos x\)
\((0\leqq x\leqq \pi)\)
\(f'(x)=1+\sin x\)
\(f''(x)=\cos x\)
増減表にまとめると、
| \(x\) | \(0\) | \(\cdots\) | \(\displaystyle \frac{1}{2}\pi\) | \(\cdots\) | \(\pi\) |
| \(f'(x)\) | \(1\) | \(+\) | \(2\) | \(+\) | \(1\) |
| \(f''(x)\) | \(1\) | \(+\) | \(0\) | \(-\) | \(-1\) |
| \(f(x)\) | \(-1\) | \(⤴\) | \(\displaystyle \frac{1}{2}\pi\) | \(↱\) | \(\pi+1\) |
したがって、
\(x=\pi\)のとき、極大値\(\pi+1\)
(3)\(f(x)=-x^4+4x^3-6x^2+4x\)
\(f'(x)=-4x^3+12x^2-12x+4=-4(x-1)^3\)
\(f''(x)=-12x^2+24x-12=-12(x-1)^2\)
増減表にまとめると、
| \(x\) | \(\cdots\) | \(1\) | \(\cdots\) |
| \(f'(x)\) | \(+\) | \(0\) | \(-\) |
| \(f''(x)\) | \(-\) | \(0\) | \(-\) |
| \(f(x)\) | \(↱\) | \(1\) | \(⤵\) |
したがって、
\(x=1\)のとき、極大値\(1\)
(4)\(f(x)=x^4-6x^2+5\)
\(f'(x)=4x^3-12x=4x(x^2-3)\)
\(f''(x)=12x^2-12=12(x+1)(x-1)\)
増減表にまとめると、
| \(x\) | \(-\sqrt{3}\) | \(\cdots\) | \(-1\) | \(\cdots\) | \(0\) | \(\cdots\) | \(1\) | \(\cdots\) | \(\sqrt{3}\) |
| \(f'(x)\) | \(0\) | \(+\) | \(+\) | \(+\) | \(0\) | \(-\) | \(-\) | \(-\) | \(0\) |
| \(f''(x)\) | \(+\) | \(+\) | \(0\) | \(-\) | \(-\) | \(-\) | \(0\) | \(+\) | \(+\) |
| \(f(x)\) | \(-4\) | \(⤴\) | \(0\) | \(↱\) | \(5\) | \(⤵\) | \(0\) | \(⤷\) | \(-4\) |
したがって、
\(x=0\)のとき、極大値\(5\)
\(x=\pm\sqrt{3}\)のとき、極小値\(-4\)