if a parabola is defined by f(x)=ax+bx+c,where a straight line is defined by g(x)=-1/2x.then Df=R,Rf=(-2,infinity) and b/2a=2
determine a,b and c and write down the equation of f.
To determine the values of a, b, and c and write down the equation for the parabola, we need to use the given information and conditions.
1. We are given that the parabola is defined by f(x) = ax^2 + bx + c. However, in your question, it seems like there is a mistake in the equation provided. The equation should be f(x) = ax^2 + bx + c, not f(x) = ax + bx + c.
2. The given condition states that the derivative of the parabola, f'(x), is represented by a straight line g(x) = -1/2x. This implies that the coefficient of the x term in the parabola's equation, which is 2a, must be equal to the slope of the line, which is -1/2.
Therefore, we can write the equation as:
2a = -1/2
3. Solving for a:
Dividing both sides of the equation by 2:
a = (-1/2) / 2
a = -1/4
4. Now that we have the value of a, we can determine b by using the fact that b/2a = 2. Substituting the value of a we found in the previous step:
b/2(-1/4) = 2
Multiply both sides by -1 to simplify the equation:
b/(-2/4) = -2
Simplifying the left side by multiplying b by -2/4:
b/(-1/2) = -2
b = (-1/2) * (-2)
b = 1
5. Finally, we need to determine the value of c. To do this, we can use the fact that Df = R and Rf = (-2, ∞). Since the discriminant Df = b^2 - 4ac, and Df = R, it means that the discriminant is positive.
Therefore, we can write:
b^2 - 4ac > 0
Substituting the values we found for a and b:
(1)^2 - 4(-1/4)c > 0
Simplifying the equation:
1 + c > 0
Now, we can see that c must be greater than -1 for the equation to satisfy the given conditions.
Hence, the values for a, b, and c are:
a = -1/4
b = 1
c > -1
The equation for the parabola is:
f(x) = (-1/4)x^2 + x + c, where c > -1.