Question

Connecting \(f\) and \(f^{\prime \prime}\) Sketch a smooth curve \(y=f(x)\) through the origin with the properties that \(f^{\prime \prime}(x)<0\) for \(x<0\) and \(f^{\prime \prime}(x)>0\) for \(x>0\)

Short answer

The equation \(y=x^3\) satisfies the required conditions. The graph of the function is concave down for \(x<0\) and concave up for \(x>0\), and passes through the origin.

Step-by-step solution

Identify a suitable function

We start by selecting a function. A good candidate function is \(y=x^3\). This function passes through the origin and changes concavity at \(x=0\).

Find the second derivative

We calculate the second derivative of the chosen function. The derivative of \(y=x^3\) is \(f'(x)=3x^2\). The second derivative, \(f''(x)\), is the derivative of \(f'(x)\), which evaluates to \(6x\).

Prove the criteria

Let's examine the concavity of the function based on \(f''(x)\). When \(x<0\), \(f''(x)<0\), this indicates that the function is concave down for all \(x<0\). When \(x>0\), \(f''(x)>0\), this shows that the function is concave up for all \(x>0\). This matches the criteria specified in the problem.

Sketch the function

Draw the graph of \(y=x^3\). The function originates at the origin, is concave down when \(x<0\) and concave up when \(x>0\).

Key concepts

Second Derivative Test

The second derivative test is a handy tool for identifying the concavity of a graph at a specific point and determining the nature of critical points. This comes in exceptionally helpful when trying to understand the behavior of a function based on its derivative.

Consider a function, denoted by the symbol \( f(x) \), and its second derivative \( f''(x) \). If \( f''(x) > 0 \) at a particular point, the graph of the function is concave up at that point. Visually, this can be compared to the way a bowl is shaped upward, where any line segment between two points on the graph will lie above or on the graph itself.

Conversely, if \( f''(x) < 0 \), the function is concave down at that point, synonymous with the shape of an umbrella, i.e., any line segment between two points on the graph will lie below the graph. If the second derivative is zero, it could signal a point of inflection, where the concavity changes. However, additional tests are needed to confirm a point of inflection, as the \( f''(x) \) being zero isn't a guarantee of that.

In our exercise, when we apply the second derivative test to the function \( y = x^3 \), we find \( f''(x) = 6x \). Here, the sign of \( x \) dictates the concavity: positive \( x \) leads to a positive second derivative (concave up), and negative \( x \) gives a negative second derivative (concave down).

Concave Up and Down

Understanding whether a function's graph is concave up or concave down is crucial in analyzing the function's behavior. The concavity tells us how the function is bending and thus, indicates potential maxima, minima or inflection points.

When we say a function is concave up, imagine that the curve is capable of holding water, like a cup. Mathematically, this will happen when its second derivative, \( f''(x) \), is greater than zero. For functions with this property, the tangent lines lie below the curve, and they tend to exhibit a minimum point at any stationary points.

On the other hand, a function is concave down when it is shaped like the outer surface of a dome. Mathematically, this coincides with a second derivative that is less than zero. The tangent lines to a concave down curve lie above the curve, and stationary points found in this region are likely to be maximums.

An easy way to remember is: if the curve looks like a 'smile' (\textbackslash{}), it's concave up; if it looks like a 'frown' (/), it's concave down. In the context of our exercise, for \( x<0 \), \( y=x^3 \) is concave down \( \left(f''(x) < 0\right) \) and for \( x>0 \) it's concave up \( \left(f''(x) > 0\right) \).

Sketching Curves

Sketching curves is an integral part of understanding the geometry of functions. A precise graph can visually demonstrate the nature of a function, including intervals of increase or decrease, local maximums and minimums, points of inflection, and concavity.

To sketch a curve accurately, one generally follows a series of steps involving finding critical points, analyzing the first and second derivatives, and identifying asymptotes if applicable. The first derivative provides information about the rate of change and thus helps to locate where the function is increasing or decreasing. The second derivative, which we focus on more intensely in our pain exercise, reveals the concavity.

Once we have determined where the function is concave up and down, we use this information to sketch the graph. As seen in our exercise with the function \( y = x^3 \), we know that the graph must pass through the origin (since when \( x = 0 \), \( y = 0 \) also), be concave down when \( x < 0 \) and be concave up when \( x > 0 \). Drawing this out, we can expect a smooth curve resembling the shape of a water slide, transitioning smoothly through the point of inflection at the origin. Such visual techniques are incredibly powerful for conceptualizing mathematical phenomena and making abstract concepts more concrete.