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
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
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)
To find the coordinates of the point of inflection on a curve, we need to find the second derivative of the function and then solve for the x-coordinate of the inflection point. Once we have the x-coordinate, we can substitute it back into the original equation to find the y-coordinate.
Let's start with curve (a): y = (x-2)²(x-7).
1. Find the first derivative:
y' = 2(x-2)(x-7) + (x-2)²
2. Simplify the equation for the first derivative:
y' = 2(x² - 9x + 14) + (x² - 4x + 4)
= 2x² - 18x + 28 + x² - 4x + 4
= 3x² - 22x + 32
3. Find the second derivative:
y'' = d/dx(3x² - 22x + 32)
= 6x - 22
4. Set the second derivative equal to zero and solve for x:
6x - 22 = 0
6x = 22
x = 22/6
x = 11/3
5. Substitute the x-coordinate back into the original equation to find the y-coordinate:
y = (11/3 - 2)²(11/3 - 7)
= (-1/3)²(-14/3)
= 1/9 * (-14/3)
= -14/27
Therefore, the coordinates of the point of inflection on curve (a) are (11/3, -14/27).
Now let's move on to curve (b): y = 4x³ + 3x² - 18x - 9.
1. Find the first derivative:
y' = 12x² + 6x - 18
2. Find the second derivative:
y'' = d/dx(12x² + 6x - 18)
= 24x + 6
3. Set the second derivative equal to zero and solve for x:
24x + 6 = 0
24x = -6
x = -6/24
x = -1/4
4. Substitute the x-coordinate back into the original equation to find the y-coordinate:
y = 4(-1/4)³ + 3(-1/4)² - 18(-1/4) - 9
= -1/4 + 3/16 + 9/4 - 9
= -1/4 + 3/16 + 36/16 - 9
= -4/16 + 3/16 + 36/16 - 144/16
= -105/16
Therefore, the coordinates of the point of inflection on curve (b) are (-1/4, -105/16).
Based on the calculations, the coordinates you provided for curve (a) are incorrect. The correct coordinates for curve (a) are (11/3, -14/27).