Problem 53 - A Complete Resource Book in Mathematics for JEE Main

Question

Let \(a, b, c \in R, a>0\) and function \(f: R \rightarrow R\) be defined by \(f(x)=a x^{2}+b x+c\) Statement 1: \(b^{2}<4 a c \Rightarrow f(x)>0\), for every value of \(x\). Statement 2: \(f\) is strictly decreasing in the interval \(\left(-\infty, \frac{-b}{2 a}\right)\) and strictly increasing in the interval \(\left(\frac{-b}{2 a}, \infty\right) .\) (A) Statement- 1 istrue, Statement- 2 is true, Statement-2 is a correct explanation for Statement-1. (B) Statement- 1 is true, Statement- 2 is true, Statement-2 is not a correct explanation for Statement-1. (C) Statement- 1 is true, Statement- 2 is false. (D) Statement- 1 is false, Statement- 2 is true.

Step-by-Step Solution

Answer

(B) Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation for Statement 1.

Step 1: Understanding Statement 1

Statement 1 claims that if \(b^2 < 4ac\), then the quadratic function \(f(x) = ax^2 + bx + c\) is positive for all \(x\). This condition resembles part of the discriminant of the quadratic formula \(b^2 - 4ac\) for the quadratic equation \(ax^2 + bx + c = 0\). If \(b^2 < 4ac\), the discriminant is negative, implying no real roots and, hence, the parabola does not intersect the x-axis. Since \(a > 0\), the parabola opens upwards and is always above the x-axis. Therefore, Statement 1 is true.

Step 2: Analyzing Statement 2

Statement 2 describes the behavior of the function \(f(x)\) concerning its increasing and decreasing intervals. The vertex form of a parabola gives that the vertex \(x = \frac{-b}{2a}\) is where the function changes from decreasing to increasing when \(a > 0\). To the left of \(x = \frac{-b}{2a}\), the function is decreasing, and to the right, it is increasing, because the vertex is the minimum point. This makes Statement 2 true.

Step 3: Connecting Statements

Even though both statements are true individually, Statement 2 describes the geometric properties of the parabola related to its vertex and symmetry. It does not directly explain or involve the condition \(b^2 < 4ac\), which relates to the discriminant and the fact the parabola does not intersect the x-axis. Thus, Statement 2 does not explain Statement 1.

Step 4: Concluding the Solution

As both statements are true, but Statement 2 is not a correct explanation for Statement 1, the answer is option (B).

Key Concepts

Discriminant

When dealing with quadratic functions, one key concept is the discriminant. The discriminant is a part of the quadratic formula and is represented as \(b^2 - 4ac\). This value is crucial because it tells us about the nature of the roots of the quadratic equation. Here is how it works:

• If the discriminant \(b^2 - 4ac\) is positive, the quadratic equation has two distinct real roots. This means the parabola crosses the x-axis at two points.
• If the discriminant is zero, there is exactly one real root, suggesting a "touch" or vertex on the x-axis.
• Finally, if the discriminant is negative, there are no real roots, which means the quadratic function does not intersect the x-axis at all. This is the case stated in Statement 1 of the exercise, where \(b^2 < 4ac\). This condition implies the parabola opens upwards and stays entirely above the x-axis if \(a > 0\), meaning it is always positive.

Understanding the discriminant helps predict how the graph of a quadratic function looks in terms of its interaction with the x-axis, highlighting whether there are real, distinct roots or none at all.

Vertex of a Parabola

The vertex of a parabola is a significant point because it represents the maximum or minimum of the quadratic function. For a function given by \(f(x) = ax^2 + bx + c\), the x-coordinate of the vertex can be found using the formula \(x = \frac{-b}{2a}\). Here's why the vertex is important:

• For a parabola that opens upwards (i.e., \(a > 0\)), the vertex is the lowest point, also known as the minimum point.
• Conversely, for a parabola that opens downward (i.e., \(a < 0\)), the vertex is the highest point, which is the maximum.

The vertex not only indicates the extremum of the function but also divides the parabola into two symmetrical halves. Identifying this point allows us to determine where the parabola switches from decreasing to increasing, which aligns with the behavior described in Statement 2. Thus, \(f(x)\) is decreasing on the interval \((-\infty, \frac{-b}{2a})\) and increasing on \((\frac{-b}{2a}, \infty)\), marking this vertex as a pivotal feature in analyzing quadratic functions.

Increasing and Decreasing Intervals

Quadratic functions demonstrate a predictable pattern in terms of increasing and decreasing intervals, which is largely dictated by the position of the vertex. The vertex tells us where the parabola starts to change direction. Understanding this helps us analyze the following:

• To the left of the vertex (before \(x = \frac{-b}{2a}\)), in the interval \((-\infty, \frac{-b}{2a})\), the function is decreasing when \(a > 0\). This means that as the x-values increase towards \(x = \frac{-b}{2a}\), the function values decrease.
• After the vertex, in the interval \((\frac{-b}{2a}, \infty)\), the function starts to increase. This is because \(a > 0\) ensures the parabola opens upwards, leading the function values to rise as x extends beyond the vertex.

These intervals are critical when studying the graph of quadratic functions because they highlight how the parabola moves along the y-axis as it progresses along the x-axis. Understanding this symmetry and behavior provides insights into the overall shape and features of the quadratic graph.