Question

Consider the general parabola described by the function $f(x)=a{x}^2+bx+c.$ For what values of $a,b,$ and $c$ is $f$ concave up? For what values of $a,b,$ and $c$ is $f$ concave down?

Short answer

Answer: For the parabola to be concave up, $a>0$. For the parabola to be concave down, $a<0$. The values of $b$ and $c$ do not affect the concavity.

Step-by-step solution

Find the first derivative of the function

Find the first derivative of the function $f(x)=a{x}^2+bx+c$ with respect to $x$. Use the power rule: $(a{x}^n{)}^{\prime }=na{x}^{n-1}$. So, $$f^{\prime }(x)=(a{x}^2{)}^{\prime }+(bx{)}^{\prime }+(c{)}^{\prime }=2ax+b$$

Find the second derivative of the function

Now, we need to find the second derivative of the function $f(x)$. We already have the first derivative as $f^{\prime }(x)=2ax+b$. Apply the power rule again to find the second derivative: $$f^{″}(x)=(2ax{)}^{\prime }+(b{)}^{\prime }=2a$$

Determine the concavity conditions

The concavity of a function is determined by the sign of its second derivative. If the second derivative is positive, the function is concave up. If the second derivative is negative, the function is concave down.

Find the values of $a,b$, and $c$ for concave up and concave down

Since the second derivative of $f(x)$ is $f^{″}(x)=2a$, we have the following conditions for concavity: - Concave up: $f^{″}(x)>0\Rightarrow 2a>0\Rightarrow a>0$ - Concave down: $f^{″}(x)<0\Rightarrow 2a<0\Rightarrow a<0$ In both cases, the value of $b$ and $c$ do not affect the concavity of the function. Therefore, we have the following: - For the parabola to be concave up, $a>0$ - For the parabola to be concave down, $a<0$ The values of $b$ and $c$ have no impact on the concavity of the function.

Key concepts

Second Derivative Test

When exploring the curvature of functions, the second derivative test is a reliable technique. It assesses the concavity of a function at a particular point by examining the second derivative at that point. The key is to find the second derivative, often denoted as $f^{″}(x)$, and observe its sign.
If $f^{″}(x)>0$, the function is concave up at that point, which visually appears as a 'smile' on the graph. Think of it as a bowl that can hold water. Conversely, if $f^{″}(x)<0$, the function is concave down, resembling a 'frown' or an upside-down bowl. It's critical to note that the second derivative test doesn't just determine concavity; it also helps in identifying inflection points, which are points where the concavity changes.

Parabolic Functions

Parabolic functions, characterized by their U-shaped graphs, are quadratic functions typically expressed in the form $f(x)=a{x}^2+bx+c$. The value of $a$ plays a pivotal role in determining the direction the parabola opens: upwards if $a>0$ and downwards if $a<0$.
The coefficients $b$ and $c$ affect the location and orientation but do not influence the parabola's concavity. Parabolas represent many real-world scenarios, such as projectile motions or the shape of satellite dishes, making them fundamental in physics and engineering. Understanding their behavior is crucial, and the concavity provides insight into their maximum or minimum values.

Concave Up and Down

The visual intuition for concavity helps understand the behavior of a function. If a function is concave up, any segment connecting two points on the function's graph will lie below or on the graph; this indicates a region where the function is increasing at an increasing rate.
On the flip side, when a function is concave down, any segment between two points on the graph lies above or on the graph, signifying a region where the function is increasing at a decreasing rate or decreasing at an increasing rate. In the case of a parabolic function, concavity is solely dictated by the leading coefficient $a$, independent of $b$ or $c$, making it a simple yet illustrative example.

Power Rule Derivation

One of the fundamental tools in calculus is the power rule for differentiation. It states that given a function $f(x)=x^n$, the derivative of that function with respect to $x$ is $f^{\prime }(x)=n{x}^{n-1}$.
In our context of quadratic functions, we apply the power rule twice: first, to find the slope of the tangent line at any point on the curve $f^{\prime }(x)=2ax+b$, and next to find the second derivative which indicates concavity $f^{″}(x)=2a$. The occurrence of the constant $2a$ after deriving $f^{\prime }(x)$ confirms that the concavity of a parabolic function depends only on the coefficient $a$, as it remains after the derivatives are taken. This ease of applying the power rule to parabolic functions builds a solid foundation for understanding more complex differentiation.