dy/dx = (x-2)^2+(x-7)2(x-2)
d^2y/dx^2 = 2(x-2) +2(x-7)+2(x-2)
= 2x - 4 + 2x - 14 + 2x - 4
= 6 x -22
zero at x = 11/3
find the point of coordinates of the point of inflexion on the curves
(a):y=(x-2)²(x-7)
(b) y=4x^3+3x²-18x-9
plz i tried my best and i got(11/6, -31/216) but keep saying am wrong
2 answers
a)
I suggest expanding this rather than using the product rule
y = (x^2 - 4x + 4)(x-7)
= x^3 - 11x^2 +32x - 28
y ' = 3x^2 - 22x +32
y '' = 6x - 22
= 0 for a point of inflection
x = 22/6 = 11/3 <------ there is your error
y = (11/3-2)^2 (11/3-7)
= (25/9)(-10/3) = -250/27
point of inflection is (11/3 , -250/27)
b) y = 4x^3+3x²-18x-9
y ' = 12x^2 + 6x - 18
y '' = 24x + 6
at the point of inflection , y '' = 0
24x = -6
x = -6/24 = -1/4
y = 4(-1/4)^3 + 3(-1/4)^2 - 18(-1/4) - 9
= -1/16 + 3/16 + 9/2 - 9
= -35/8
the point of inflection is (-1/4 , -35/8)
I suggest expanding this rather than using the product rule
y = (x^2 - 4x + 4)(x-7)
= x^3 - 11x^2 +32x - 28
y ' = 3x^2 - 22x +32
y '' = 6x - 22
= 0 for a point of inflection
x = 22/6 = 11/3 <------ there is your error
y = (11/3-2)^2 (11/3-7)
= (25/9)(-10/3) = -250/27
point of inflection is (11/3 , -250/27)
b) y = 4x^3+3x²-18x-9
y ' = 12x^2 + 6x - 18
y '' = 24x + 6
at the point of inflection , y '' = 0
24x = -6
x = -6/24 = -1/4
y = 4(-1/4)^3 + 3(-1/4)^2 - 18(-1/4) - 9
= -1/16 + 3/16 + 9/2 - 9
= -35/8
the point of inflection is (-1/4 , -35/8)