91Ó°ÊÓ

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\). They are called quadratic because the highest degree of the variable \(x\) is 2. The solutions to these equations are known as the roots, and they tell us where the graph of the equation crosses the x-axis.
To find the roots of a quadratic equation, you can use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula relies on the discriminant \(b^2 - 4ac\) to determine the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If it is zero, there is exactly one real root (a repeated root).
• If the discriminant is negative, there are no real roots, but there are two complex roots.

In solving quadratic equations, it's important to consider the interval or range in which you need the roots to exist, as demonstrated in the given exercise. This approach helps determine whether a solution fits within certain parameters for practical applications.

Continuity and differentiability are fundamental concepts in differential calculus. They help describe the behavior of functions and their graphs.
**Continuity**: A function is said to be continuous at a point if there is no interruption in its graph at that point. For a function \(f(x)\) to be continuous at \(x = c\), three conditions must be satisfied:

• \(f(c)\) is defined.
• The limit of \(f(x)\) as \(x\) approaches \(c\) exists.
• The limit of \(f(x)\) as \(x\) approaches \(c\) is equal to \(f(c)\).

**Differentiability**: If a function is continuous and smooth (no sharp corners or cusps), it is differentiable at that point. Differentiability implies the existence of a derivative, which represents the function's rate of change.
Polynomials, like \(f(x) = 3x^2 + 4x + b\), are both continuous and differentiable everywhere, including within the interval \((0,1)\), as noted in the exercise. This ensures smooth curves without breaks or sharp turns.

Polynomials are expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication. A general polynomial of degree \(n\) can be written as:\[ a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0 \]The degree of the polynomial is the highest power of \(x\) with a non-zero coefficient.
Polynomials have several key characteristics:

• They are continuous everywhere, meaning their graphs are unbroken curves.
• They are differentiable, ensuring they have a smooth tangent line at every point.
• The degree of the polynomial determines the number of roots or intersections with the x-axis.

The behavior of polynomials is predictable, making them essential tools in calculus for modeling and solving real-world problems. In the given exercise, understanding the nature of the polynomial function helps determine its continuity and differentiability, which are crucial for validating statement II.