schneider15的部落格

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 1.f(x)=3x^4+8x^3+1=0 => x=-0.539, -2.65

f'(x)=12x^3+24x^2=0 => x=0,0,-2

f"(x)=36x^2+48x

f"(0)=0 => 拐點.斜率=0(水平線)

f"(-2)=144-96=48>0 => 極小點

f(-2)=-15, f(0)=1

曲線類似拋物線.開口向上.極小點=(-2,-15).拐點=(0,1).

通過水平軸=(-5.39,0) & (-2.65,0)

x<=-2 => 遞減曲線

2<x<0 => 遞增曲線

x=0 => 持平

x>0 => 遞增曲線

2.f(x)=x^(1/3)-2x^(1/3)=-x^(1/3) => f(0)=0=min

f'(x)=-1/3*x(-4/3)=0 => x=0

f"(x)=(4/9)*x(-7/3)=(4/9)/x(7/3) => f"(0)→∞ => f(0)=min

3.f(x)=x+2/x^2 => f(2^(2/3))=2.3811=min

f'(x)=1-4/x^3=0 => x=4^(1/3)=2^(2/3)

f"(x)=12/x^4 => f(xn)=12/4^(4/3)>0 => min

4.f(x)=x^(4/5)+4x^(-1/5) => f(1)=1+4=5=min

f'(x)=4/5*x^(-1/5)-4/5*x^(-6/5)=0 => x=1

f(x)"=-4/25*x(-6/5)+24/25*x^(-11/5) => f(1)=20/25=4/5>0 => min

5.f(x)=(x-2)/(x+2)

Limit<x→-2,Left>f(x)=∞......(1)

Limit<x→-∞>f(x)=0...........(2)

Limit<x→-2,right>f(x)=-∞....(3)

Limit<x→∞>f(x)=0............(4)

(1)&(2)=2Q, (3)&(4)=4Q

6.f(x)=x^2*e^x≧0 => f(0)=1=min

f'(x)=(x^2+2x)*e^x=0 => x=0, x=-2(不合)

f"(x)=(x^2+4x+2)*e^x => f(0)=2>0 => min

7.f(x)=Ln(x)/x≧0 => x≧0, f(e)=1/e=max

f'(x)=[1-Ln(x)]/x^2=0 => x=0(不合), x=e

f"(x)=[-3+2Ln(x)]/x^3 => f"(e)=-1/e^3<0 => max

8.1.Cost c(x)=20+200x+0.01*x^3>0

x=生產數量>0

c'(x)=200+0.03*x^2>0 => 沒有極值

8.2.Average cost: a(x)=c(x)/x=20/x+200+0.01*x^2>0 (x>0)

a'(x)=-20/x^2+0.02*x=0 => x=-10(不合)

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