First show that

dy/dx = (-3x² + 4x - 15) / (2 - 3x)²
(Use your usual quotient rule)


Now, completing the square,

-3x² + 4x - 15

=-3(x² - 4/3 x + ( (4/3)/2 )² ) -15 + 3( (4/3)/2 )²

= -3(x² - ⅔ x + (⅔)²) - 15 + 3(⅔)²

= -3(x - ⅔)² - 15 + 3(4/9)

= -3(x - ⅔)² - 13⅔


Since x ≠ ⅔,

x - ⅔ ≠ 0

(x - ⅔)² > 0 for all real x
(The square of any real non-zero term is aways positive)

Then -3(x - ⅔)² < 0

-3(x - ⅔)² - 13⅔ < -13⅔ < 0

So -3x² + 4x - 15 < 0 for all real x


Which means (-3x² + 4x - 15) / (2 - 3x)² ≠ 0 for all real x and thus dy/dx

(Note : the denominator is non-zero. We cannot have a denominator being 0 as it would make the expression undefined

x ≠ ⅔ so 2 - 3x ≠ 0 and thus (2 - 3x) ≠0)

Since dy/dx ≠ 0 for all real x, there are no stationary points since stationary points are where dy/dx = 0

Alternatively,

For -3x² + 4x - 15

Discriminant (b² - 4ac)

= 4² - 4(-3)(-15)
= 16 - 180
= -164

Since discriminant < 0 , there are no real roots for the equation -3x² + 4x - 15 = 0

This means -3x² + 4x - 15 ≠ 0 for all real x