application of differentiation

首先,我本身係用中文讀M1,我會盡量用英文做
其次,我M1都唔係話好勁,我純粹只係試下做,唔知岩唔岩
仲有,y=nx^3+mx62 我唔知你有冇打錯條式,我唔知你係想打y=nx^3+mx62 or y=nx^3+mx^2 ,所以我兩個都做
1.

y=nx^3+mx62

dy/dx=3nx^(2)+62m
let 3nx^(2)+62m =0 (因為係turning point,slop=0)
因為turning point係(1,2)
所以3n(1)^(2)+62m=0 (1)
3n+62m=0

y=nx^3+mx62
2=n(1)^3)+m(1)62
2=n+62m (2)

3n+62m=0 (1)
n+62m=0 (2)
so, n= -1, m= 3/62
so,y= -x^(3)+3x


y=nx^3+mx^2
dy/dx=3n^(2)+2mx
let 3xn^(2)+2mx=0 (因為係turning point,slop=0)
因為turning point係(1,2)
所以3n^(2)+2m=0 (1)

y=nx^3+mx^2
2=n+m (2)

3n^(2)+2m=0 (1)
n+m=2 (2)

so, n= -4, m=6
so, y=-4x^(3)+6x^(2)


2. if the curve has a maximum point and a minimum point, it mean when dy/dx=0, the equation has 2 unequal roots

y=nx^3+mx62
y= -x^(3)+3x
dy/dx= -3x^(2)+3

when dy/dx=0,
-3x^(2)+3=0
x= -1 or x= 1




y=nx^3+mx^2
y= -4x^(3)+6x^(2)
dy/dx= -12x^(2)+12x
when dy/dx=0,
-12x^(2)+12x=0
x=o or x= 1

so, if the curve has a maximum point and a minimum point, it mean when dy/dx=0, the equation has 2 unequal roots

2.
If the curve has a maximum point, a minimum point and without point(s) of inflection, when dy/dx=0, the equation has 2 unequal roots.

If the curve has a maximum point, a minimum point and one or more points of inflection, when dy/dx=0, the equation has 3 or more unequal roots.