Problem 7 - A Textbook of Differential Calculus

Question

A quadratic polynomial maps from \([-2,3]\) onto \([0,3]\) and touches \(x\)-axis at \(x=3\), then the polynomial is (a) \(\frac{3}{16}\left(x^{2}-6 x+16\right)\) (b) \(\frac{3}{25}\left(x^{2}-6 x+9\right)\) (c) \(\frac{3}{25}\left(x^{2}-6 x+16\right)\) (d) \(\frac{3}{16}\left(x^{2}-6 x+9\right)\)

Step-by-Step Solution

Answer

The polynomial is (b) \(\frac{3}{25}(x^2 - 6x + 9)\).

Step 1: Understanding the Problem

The quadratic polynomial is defined to map from an interval \([-2,3]\) onto \([0,3]\)\. It touches the \(x\)-axis at \(x=3\)\, meaning the polynomial is zero at \(x=3\) with multiplicity 2 (as it touches, or is tangent to, the \(x\)-axis)\. We need to find the polynomial in the given options that satisfies these conditions\.

Step 2: Formulating the Polynomial

Since the polynomial touches the \(x\)-axis at x=3, let it be \((x-3)^2\)\. Hence, the polynomial can be written as \(f(x) = k(x-3)^2\)\, where \(k\) is a constant to adjust the range from \([-2,3]\) to \([0,3]\)\.

Step 3: Determining \\(k\\)

At the boundary \(x = -2\)\, \(f(x) \) should be \(0\)\. We calculate: \(f(-2) = k((-2)-3)^2 = 25k\)\. For \(f(3) = k(0)^2 = 0\), and since \(f(3) = 0\), the function works at \(x=3\)\. Therefore, we need \(25k = 3\) at the top of the range \([0,3]\)\, which gives \(k = \frac{3}{25}\)\.

Step 4: Form of the Polynomial

Substitute \(k = \frac{3}{25}\) into the polynomial to get \(f(x) = \frac{3}{25}(x-3)^2\)\.

Step 5: Simplifying \\( (x-3)^2 \\)

We expand \( (x-3)^2 \) to \( x^2 - 6x + 9 \)\. Therefore, the polynomial is \( f(x) = \frac{3}{25}(x^2 - 6x + 9) \)\.

Step 6: Matching with Options

We compare \( f(x) = \frac{3}{25}(x^2 - 6x + 9) \) with the given options to find that it matches option (b)\.

Key Concepts

Polynomial Mapping

Polynomial mapping is a concept where a polynomial function relates two sets of numbers: inputs (domain) and outputs (range). In this exercise, we are dealing with a quadratic polynomial mapping that transforms the interval

• From: [-2, 3] (the domain, input values for x)
• To: [0, 3] (the range, output values of the polynomial)

Mapping ensures that every input corresponds to an appropriate output within the specified range. This is done by adjusting constants such as the coefficient in our quadratic expression. When a polynomial "maps" its domain to its range correctly, all points in the input interval will satisfy the equation and produce output values within the range. In this context, the focus was to ensure that after defining the polynomial, the values computed for the interval the input precisely fit into the output interval.

Quadratic Functions

Quadratic functions describe a special type of polynomial that is characterized by the term with the power of two, written as \(ax^2 + bx + c\). They produce a parabolic curve when graphed, either opening upwards or downwards, depending on the sign of coefficients. Quadratic functions are very significant in this exercise.
For the given exercise, the polynomial is assumed to be in the form \((x-3)^2\), because it touches the x-axis at \(x=3\). Here, the quadratic term ensures the curve's lowest or highest point touches the x-axis, which is necessary to match the mapping condition.
Quadratics are defined by:

• Vertex: It's the point where the graph turns; in this case, \(x=3\) is the vertex.
• Axis of Symmetry: A vertical line that divides the graph into two symmetrical halves, passing through the vertex.

In our problem, adjusting the function with a constant \(k\) maintains the necessary range and ensures symmetry about the vertex.

Roots of Polynomials

Roots, or zeros, of polynomials are values of \(x\) for which the polynomial equals zero. For quadratic functions, the roots are points at which the curve intersects the x-axis. However, sometimes instead of crossing, the curve only touches the axis at the root and moves away—this is known as a root with "multiplicity more than one."
In this exercise, the polynomial has a root at \(x=3\), and because it merely touches the x-axis there, the root has multiplicity 2. This means \((x-3)^2\) creates a tangent to the x-axis at this point.

• Multiplicity: Multiplicity refers to the number of times a particular root appears. A higher multiplicity creates a flatter or more pronounced touch at the axis, instead of passing through.
• Effect on Graph: For a root with even multiplicity, like our example, the graph touches the axis and turns back. Odd multiplicities result in the curve crossing through the axis.

Understanding the roots and their multiplicities is crucial for accurately formulating and verifying practical polynomial mappings as we did in this exercise.