Question

In \(\mathrm{a}+\mathrm{b}+\mathrm{c}=0\), than the equation \(3 \mathrm{ax}^{2}+2 \mathrm{bx}+\mathrm{c}=0\) has, in the interval \((0,1)\) (a) at least one root (b) at most one root (c) no root (d) Exactly one root exist.

Short answer

In the given quadratic equation \(3ax^2 + 2bx + c = 0\) with \(a + b + c = 0\), after analyzing its critical points and examining its behavior in the interval (0,1), we can conclude that exactly one root exists in that interval. Thus, the correct option is (d).

Step-by-step solution

Rewrite the given equations

We have two equations: \[a + b + c = 0\] and \[3ax^2 + 2bx + c = 0\] Rewrite the quadratic equation to better study its properties: \[f(x) = 3ax^2 + 2bx + c\]

Find the critical points using the first derivative

Compute the first derivative of f(x): \[f'(x) = \frac{d}{dx} (3ax^2 + 2bx + c)\] \[f'(x) = 6ax + 2b\] Critical points occur when the first derivative is equal to 0. Solve for x: \[6ax + 2b = 0\] \[x = -\frac{b}{3a}\]

Investigate the critical point

Let's analyze the critical point -\(\frac{b}{3a}\) in respect to the interval (0,1). Since a, b, and c are not all zero, there are three possibilities: 1. a, b > 0 and c = -a - b < 0 2. a, b < 0 and c = -a - b > 0 3. a > 0 > b and c = -a - b For the first case, a, b > 0, we have the critical point -\(\frac{b}{3a}\) < 0 and f(1) = \(-a - a + 2b = b - a > 0\), which makes the curve positive outside of the interval (0,1). For this case, there's no root (`c`). For the second case, we have a and b < 0, the critical point -\(\frac{b}{3a}\) > 0 and f(1) = \(-a - a + 2b = b - a < 0\), which makes the curve negative outside of the interval (0,1). For this case, there's no root (`c`). For the third case, we have a > 0 > b, the critical point -\(\frac{b}{3a}\) lies inside the interval (0,1). Here, f(0) = c and f(1) = b - a != 0. Also, since -\(\frac{b}{3a}\) is in (0,1), we have exactly one root in the interval (`d`).

Select the correct option

Based on our analysis in Step 3, options (a), (b), and (c) are incorrect, while option (d) is: (d) Exactly one root exists.

Key concepts

Critical Points

Critical points in a function are essential spots where the derivative equals zero or is undefined. This concept is crucial because it helps identify where a function may have a local maximum, minimum, or neither. Specifically, for the quadratic function we are analyzing, \[ f(x) = 3ax^2 + 2bx + c \]we find the critical points by setting the first derivative to zero. Calculating the derivative, we have:\[ f'(x) = 6ax + 2b \]Set this to zero to find the critical points: \[ 6ax + 2b = 0 \]. By solving for \( x \), we get:\[ x = -\frac{b}{3a} \]. These critical points are crucial in determining how the function behaves within specific intervals. When this result is evaluated in terms of the problem's interval, (0,1), it helps determine the existence or non-existence of roots within that interval.

Derivative Test

The derivative test is a useful tool to analyze a function's behavior around its critical points. In our quadratic equation, \( f(x) = 3ax^2 + 2bx + c \), the first derivative is used to identify these critical points:\[ f'(x) = 6ax + 2b \].Once we find the critical point, in this case, \[ x = -\frac{b}{3a} \], we can use the derivative test to conclude the function's behavior at that point.The test helps us determine whether the critical point is a local maximum, minimum, or a point of inflection:

• If the first derivative changes sign from positive to negative at a critical point, the function has a local maximum there.
• If the first derivative changes sign from negative to positive, the function has a local minimum.
• If there's no sign change, it might be a point of inflection.

Applying this to the context of our quadratic relationship, the point \(-\frac{b}{3a}\) and its placement relative to the interval (0,1) guides us to discern the number of roots in this interval.

Roots in Intervals

Roots of a function are the values of \( x \) that make the function equal to zero. With quadratic equations like \( 3ax^2 + 2bx + c = 0 \), identifying roots within certain intervals is crucial in solving related problems.In our problem, we are interested in the interval (0,1). The key to finding the roots within this interval is understanding how the function changes within the interval and how its critical point affects this.For our function, by assessing the signs and behaviors at critical points and endpoints of the interval (0 and 1), we can determine:

• Whether the graph of the function crosses or touches the x-axis within (0,1).
• Whether the critical point \( -\frac{b}{3a} \) lies inside the interval.
• The corresponding value of the function at those critical points and endpoints.

In our example, these assessments show it's possible to determine that exactly one root exists in the interval (0,1), based on the function's behavior at this critical juncture, reinforcing the solution choice of "d) Exactly one root exists."