Question

Which of the following quadratic functions are strictly convex? (a) \(y=9 x^{2}-4 x+8\) (c) \(u=9-2 x^{2}\) (b) \(w=-3 x^{2}+39\) \((d) v=8-5 x+x^{2}\)

Short answer

Functions (a) and (d) are strictly convex.

Step-by-step solution

Understand Convexity

A quadratic function is convex if the coefficient of the squared term is positive. Specifically, in the form \(ax^2 + bx + c\), the function is strictly convex if \(a > 0\).

Analyze Option (a)

For the function \(y=9x^2-4x+8\), identify the coefficient of \(x^2\), which is 9. Since 9 is positive, this function is strictly convex.

Analyze Option (b)

The function is \(w=-3x^2+39\). The coefficient of \(x^2\) is -3, which is negative. Therefore, this function is not convex.

Analyze Option (c)

For the function \(u=9-2x^2\), the coefficient of \(x^2\) is -2. This is negative, so the function is not convex.

Analyze Option (d)

The function is \(v=8-5x+x^2\). The coefficient of \(x^2\) is 1, which is positive. Hence, this function is strictly convex.

Summarize Findings

The functions from options (a) and (d) are strictly convex because they have positive coefficients for the \(x^2\) term.

Key concepts

Convexity

Convexity in mathematics, especially in the context of quadratic functions, is a property that informs us about the curve's shape. It gives insight into how a function behaves over a given interval. When discussing convexity, we are typically concerned with whether the graph of a function will open upwards or downwards. This is important because it reveals how the function's outputs (or values) change as the inputs increase or decrease.

For a quadratic function expressed in the standard form \(ax^2 + bx + c\):

• If the leading coefficient \(a > 0\), the graph of the function is a parabola that opens upwards, indicating convexity. Such functions are typically referred to as being 'convex.'
• If \(a < 0\), the parabola opens downwards, indicating concavity.

Convex functions are particularly beneficial in optimization problems. When a function is convex, any local minimum is also a global minimum, which simplifies the process of finding the function's optimal values.

Coefficient of Quadratic Term

The coefficient of the quadratic term \(a\) in a quadratic function \(ax^2 + bx + c\) plays a crucial role in determining the function's nature. It directly affects the graph's orientation and hence the function's convexity or concavity. Understanding the role of this coefficient helps in quickly assessing the type of function we are dealing with.

Here's how you can interpret the coefficient of the quadratic term:

• Positive Coefficient (\(a > 0\)): When the coefficient is positive, the parabola opens upwards, and the function is reported as convex. This indicates that the function values rise on either side of the vertex, creating a 'U' shape.
• Negative Coefficient (\(a < 0\)): Conversely, if the coefficient is negative, the parabola opens downwards, demonstrating a 'n' shape which indicates concavity.

Typically, in the context of strictly convex functions, a positive coefficient is a clear indicator that the function will feature properties utilized heavily in fields like economics, optimization, and engineering.

Strictly Convex Functions

Strictly convex functions are a sub-category of convex functions in which the curve is consistently curved upwards, without flat regions. These functions have a unique characteristic that sets them apart—they have a single point of minimum, precisely at their vertex if considering a quadratic form.

In practical terms, a strictly convex function exhibits:

• Positive Coefficient Requirement: The hallmark of being strictly convex is having a strictly positive coefficient \(a > 0\). This ensures the parabola is opening upwards without any flat or horizontal segments.
• Unique Minimum: Strict convexity means that, aside from the continuous rising nature, there is precisely one minimum point, making such functions critical in finding binding constraints in optimization.

To determine if a quadratic function is strictly convex, one can simply inspect the sign of the coefficient of the quadratic term. A positive sign confirms the strict convexity, while any flat portion or negative value of \(a\) would rule it out.