Problem 7 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

Let \(f(x)=x^{2}+4 x+1\), then (a) \(f(x)>0\) for all \(x\) (b) \(f(x)>1\) when \(x \geq 0\) (c) \(f(x) \geq 1\) when \(x \leq-4\) (d) \(f(x)=f(-x)\) for all \(x\)

Step-by-Step Solution

Answer

(b) and (c) are true; (a) and (d) are false.

Step 1: Understanding the quadratic function

The function given is a quadratic function of the form \(f(x) = ax^2 + bx + c\) where \(a = 1\), \(b = 4\), and \(c = 1\). The discriminant of a quadratic, \(b^2 - 4ac\), helps determine the nature of its roots and the sign of the function.

Step 2: Analyze condition (a)

For \(f(x) > 0\) for all \(x\), the quadratic must not intersect the x-axis and must open upwards. Since \(a = 1 > 0\), the parabola opens upwards. Calculate the discriminant: \[\Delta = b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot 1 = 16 - 4 = 12.\]The discriminant is positive, meaning there are two distinct real roots. Thus, (a) is false since the function is zero at some points.

Step 3: Analyze condition (b)

Check if \(f(x) > 1\) when \(x \geq 0\). Rewrite the condition:\[x^2 + 4x + 1 > 1\]\[x^2 + 4x > 0\]Factor out \(x\):\[x(x + 4) > 0\]This inequality holds for \(x < -4\) and \(x > 0\), thus for \(x \geq 0\), \(f(x) > 1\) is true.

Step 4: Analyze condition (c)

Verify \(f(x) \geq 1\) when \(x \leq -4\). Consider:\[x^2 + 4x + 1 \geq 1\]Implying\[x^2 + 4x \geq 0\]Factoring, \(x(x + 4) \geq 0\) holds when \(x \leq -4\) or \(x \geq 0\). Hence, (c) is true.

Step 5: Analyze condition (d)

Check for symmetry: \(f(x) = f(-x)\) implies:\[x^2 + 4x + 1 = x^2 - 4x + 1\]Simplify:\[4x = -4x\]\[8x = 0\]Unless \(x = 0\), this won't hold, thus (d) is false.

Key Concepts

Discriminant of a Quadratic

The discriminant of a quadratic equation is a crucial concept for understanding the nature of the roots of the equation. For a quadratic function of the form \(f(x) = ax^2 + bx + c\), the discriminant \(\Delta\) is given by the expression \(b^2 - 4ac\). This value tells us how the parabola intersects the x-axis.

• If \(\Delta > 0\), the equation has two distinct real roots, meaning the parabola crosses the x-axis at two points.
• If \(\Delta = 0\), there is exactly one real root, and the parabola touches the x-axis at a single point (the vertex).
• If \(\Delta < 0\), there are no real roots, and the parabola does not intersect the x-axis at all.

In our exercise, the discriminant is \(12\), which is greater than zero. This indicates two distinct intercepts, meaning condition (a) that "\(f(x) > 0\) for all \(x\)" is false.
Understanding the discriminant is essential for predicting the behavior of a quadratic function.

Inequalities

Inequalities help us understand the range and behavior of quadratic functions over certain intervals. For condition (b) in our exercise, we examined the inequality \(f(x) > 1\) for \(x \geq 0\).

• To solve, we began with the expression \(x^2 + 4x + 1 > 1\), simplifying it to \(x^2 + 4x > 0\).
• By factoring, \(x(x + 4) > 0\), we determined that the inequality holds for \(x > 0\) or \(x < -4\). Therefore, when \(x \geq 0\), the condition is fulfilled, marking condition (b) as true.

Inequalities guide us in understanding where the function increases or decreases relative to specific values.

Parabola

A quadratic function forms a curve called a parabola when graphed on a coordinate plane. This shape is defined by its vertex, axis of symmetry, and direction of opening. The direction is determined by the coefficient \(a\) in the quadratic equation. If \(a > 0\), the parabola opens upwards. Conversely, if \(a < 0\), it opens downwards.

• The vertex is the topmost or bottommost point of the parabola where the function reaches a minimum or maximum value.
• The axis of symmetry is a vertical line passing through the vertex that divides the parabola into two mirror-image halves.

In our function \(f(x) = x^2 + 4x + 1\), since \(a = 1 > 0\), it confirms that the parabola opens upwards. Understanding the structure of a parabola is vital for visualizing the behavior of quadratic functions and analyzing inequalities within specific intervals.

Symmetry in Functions

Symmetry in mathematical functions indicates a balanced, mirror-like distribution about a line or point. For quadratic functions, finding symmetry is often about determining if the function is even, meaning \(f(x) = f(-x)\).
In our exercise, checking condition (d) required testing whether \(f(x) = x^2 + 4x + 1\) could equal \(f(-x) = x^2 - 4x + 1\):

• Comparing both sides shows they are equal only if the terms containing \(x\) cancel out, i.e., \(4x = -4x\).
• This leads to \(8x = 0\), which is true only if \(x = 0\).

Because this equation does not hold for all possible \(x\) values, \(f(x)\) is not fully symmetric around the y-axis. This exploration highlights the importance of symmetry in simplifying and analyzing mathematical expressions, even if only partial symmetry is present.