Final answer:
The function f(x) = 3/2x³ - 4x² + 10x has no relative extreme points.
Step-by-step explanation:
To show that the function f(x) = \frac{3}{2}x^3 - 4x^2 + 10x has no relative extreme points, we first need to understand that relative extreme points exist when the first derivative of the function equals zero and the second derivative test confirms a minimum or maximum. For f(x), we find the first derivative:
f'(x) = \frac{d}{dx}(\frac{3}{2}x^3 - 4x^2 + 10x) = \frac{9}{2}x^2 - 8x + 10
Setting the first derivative equal to zero, we get:
\frac{9}{2}x^2 - 8x + 10 = 0
This is a quadratic equation, which we would generally solve to find potential relative extreme points. However, using the discriminant b^2 - 4ac from the quadratic formula, we determine whether this equation has real solutions. Substituting a = \frac{9}{2}, b = -8, and c = 10, the discriminant is:
(-8)^2 - 4\cdot(\frac{9}{2})\cdot10 = 64 - 180 = -116
Since the discriminant is negative, the equation has no real solutions. Therefore, f'(x) = 0 has no solutions and the function f(x) has no relative extreme points within the given domain.