Problem 53 - A Complete Resource Book in Mathematics for JEE Main 2017

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 is true, Statement- 2 istrue, 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: Review Statement 1 Condition

For the quadratic function \( f(x) = ax^2 + bx + c \), where \( a > 0 \), the condition \( b^2 < 4ac \) implies the quadratic has no real roots. Since \( a > 0 \), the quadratic opens upwards, which means its vertex is the lowest point on the graph. Without real roots, the entire parabola must lie above the x-axis, ensuring \( f(x) > 0 \) for all \( x \). Thus, Statement 1 is true.

Step 2: Analyze Statement 2

The function \( f(x) = ax^2 + bx + c \) is a parabola that opens upward. The vertex of the parabola is at \( x = -\frac{b}{2a} \). To find where \( f(x) \) is increasing or decreasing, consider its derivative \( f'(x) = 2ax + b \). For \( x < -\frac{b}{2a} \), \( f'(x) < 0 \), indicating the function is decreasing. For \( x > -\frac{b}{2a} \), \( f'(x) > 0 \), indicating the function is increasing. Thus, Statement 2 is true.

Step 3: Determine if Statement 2 Explains Statement 1

Statement 1 asserts that the quadratic is always positive when \( b^2 < 4ac \), while Statement 2 describes the intervals of increase or decrease. While Statement 2 accurately describes the parabola's behavior, it does not directly explain why \( f(x) > 0 \) for all \( x \) under the condition \( b^2 < 4ac \). Therefore, Statement 2 is not a correct explanation for Statement 1.

Key Concepts

Quadratic Inequalities

Quadratic inequalities involve expressions where a quadratic function is compared to zero or another function. A typical form is ax² + bx + c > 0, where the challenge lies in determining the intervals of x that satisfy this inequality.

To solve a quadratic inequality like this, often you'll look at the roots of the corresponding equation ax² + bx + c = 0. These roots divide the number line into intervals.

By testing points from each interval, you can see where the original inequality holds true. However, there is a special case when there are no real roots. Consider b² < 4ac; here, the parabola does not intersect the x-axis because the discriminant is negative.

This means the entire quadratic curve lies either above or below the x-axis, depending on the leading coefficient, a. If a > 0, the parabola opens upwards, ensuring the entire curve stays above the x-axis.

This critical understanding helps solve quadratic inequalities more efficiently.

Vertex of a Parabola

The vertex of a parabola is a significant point because it represents the highest or lowest point on the graph, depending on its orientation.

For a quadratic function like f(x) = ax² + bx + c, this turning point can be found using the formula x = -b / (2a). This value of x gives the location of the vertex along the x-axis.

If a > 0, the parabola opens upwards, making the vertex the lowest point on the graph. Conversely, if a < 0, it opens downwards, with the vertex as the highest point.

The actual coordinates of the vertex are (–b/2a, f(–b/2a)). This knowledge is crucial because it helps determine the behavior of the quadratic function, pinpointing exactly where it changes from decreasing to increasing or vice versa on the graph.

Increasing and Decreasing Functions

Understanding where a function increases or decreases is pivotal in analyzing its behavior. For quadratic functions like f(x) = ax² + bx + c, this can be derived from its derivative.

The derivative f'(x) = 2ax + b tells us the rate of change of the function. By setting f'(x) = 0, we can find the critical point, which is also where the vertex of the parabola lies, x = -b/2a.

At points to the left of the vertex, x < -b/2a, the derivative f'(x) < 0 indicates the function is decreasing. Conversely, for x > -b/2a, f'(x) > 0, meaning the function is increasing.

This method gives a clear picture of the function's ups and downs, which is not only useful for sketching graph behaviors but also for solving optimization problems and understanding how the graph shifts.