Final answer:
For the function f(x)=2x²-3x+4, f(x) is always greater than zero for all real numbers x because the discriminant of the quadratic equation is negative, indicating there are no real roots, and since the parabola opens upwards.
Step-by-step explanation:
To determine where f(x) is greater than zero and where it is less than zero for the function f(x)=2x²-3x+4, we must analyze the function's behavior by looking at its quadratic formula.
First, we note that this is a quadratic equation in the standard form ax²+bx+c=0. In this case, a=2, b=-3, and c=4. Since a>0, the parabola opens upwards, and f(x) will be greater than zero for most values of x except possibly between the roots, if there are any real roots.
Next, we calculate the discriminant Δ=b²-4ac to determine the nature of the roots. After substituting the values in, we get Δ=(-3)²-4×2×4 which simplifies to 9-32, yielding a discriminant of -23. Since the discriminant is less than zero, there are no real roots, and therefore f(x) does not cross the x-axis and remains positive for all x.
In conclusion, for the given function f(x)=2x²-3x+4, f(x) is always greater than zero and never less than zero for all real numbers x.