/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 7 Let \(f(x)=x^{2}+4 x+1\), then ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 7

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 \(x\)

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.
04

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.
05

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.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If \((x+a)\) is a factor of both the quadratic polynomials \(x^{2}+p x+q\) and \(x^{2}+l x+m\), where \(p, q, l\) and \(m\) are constants, then which one of the following is correct? (a) \(a=(m-q) /(l-p)(l \neq p)\) (b) \(a=(m+q) /(l+p)(l \neq-p)\) (c) \(l=(m-q) /(a-p)(a \neq p)\) (d) \(p=(m-q) /(a-l)(a \neq l)\)

The number of roots of the equation \(|x|^{2}-7\) \(|x|+12=0\) is (a) 1 (b) 2 (c) 3 (d) 4

What is the value of \(x\) satisfying the equation \(16\left(\frac{a-x}{a+x}\right)^{3}=\frac{a+x}{a-x} ?\) (a) \(a / 2\) (b) \(a / 3\) (c) \(a / 4\) (d) 0

If \(x\) is real, then the maximum and minimum values of the expression \(\frac{x^{2}-3 x+4}{x^{2}+3 x+4}\) will be (a) 2,1 (b) \(5,1 / 5\) (c) \(7,1 / 7\) (d) none of these

If both roots of equations \(K\left(6 x^{2}+3\right)+r x+\) \(2 x^{2}-1=0\) and \(6 K\left(2 x^{2}+1\right)+p x+4 x^{2}-2=\) 0 are common, then prove that \(2 r-p=0\).
See all solutions

Recommended explanations on Math Textbooks

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.