Answer:
P( - [tex]\frac{1}{2}[/tex], [tex]\frac{5}{2}[/tex] )
Step-by-step explanation:
The derivative of y is the gradient of the slope.
Given
y = 2x³ + 3x² + 2, then
[tex]\frac{dy}{dx}[/tex] = 6x² + 6x , equate to - [tex]\frac{3}{2}[/tex]
6x² + 6x = - [tex]\frac{3}{2}[/tex] ( add -
6x² + 6x + [tex]\frac{3}{2}[/tex] = 0 ( multiply through by 2 )
12x² + 12x + 3 = 0 ( divide through by 3 )
4x² + 4x + 1 = 0 ← in standard form
(2x + 1)² = 0 ← in factored form
2x + 1 = 0 ⇒ 2x = - 1 ⇒ x = - [tex]\frac{1}{2}[/tex]
Substitute this value into y for corresponding y- coordinate
y = 2(- [tex]\frac{1}{2}[/tex] )³ + 3(-
= 2(- [tex]\frac{1}{8}[/tex] ) + 3([tex]\frac{1}{4}[/tex] ) + 2
= - [tex]\frac{1}{4}[/tex] + [tex]\frac{3}{4}[/tex] + 2 = [tex]\frac{5}{2}[/tex]
Thus P = ( - [tex]\frac{1}{2}[/tex], [tex]\frac{5}{2}[/tex] )
