Question

Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points. $$f(x)=2x^4+8x^3+12x^2-x-2$$

Short answer

Question: For the function $f(x)=2x^4+8x^3+12x^2-x-2$, identify the intervals in which the function is concave up, concave down, and any inflection points if they exist. Answer: The function is concave up for all values of x. There are no inflection points.

Step-by-step solution

Find the first derivative of the function.

The function we are given is $$f(x)=2x^4+8x^3+12x^2-x-2$$ To find the first derivative, $f^{\prime }(x)$, differentiate with respect to x: $$f^{\prime }(x)=\frac{d}{dx}(2x^4+8x^3+12x^2-x-2)$$ $$f^{\prime }(x)=8x^3+24x^2+24x-1$$

Find the second derivative of the function.

Now, differentiate $f^{\prime }(x)$ with respect to x to find the second derivative, $f^{″}(x)$: $$f^{″}(x)=\frac{d}{dx}(8x^3+24x^2+24x-1)$$ $$f^{″}(x)=24x^2+48x+24$$

Test for concavity and find inflection points.

Now we need to find the intervals where $f^{″}(x)$ is positive and when $f^{″}(x)$ is negative. We will also identify any inflection points, which are the points where the concavity changes. To do this, first set $f^{″}(x)$ equal to zero and solve for x: $$24x^2+48x+24=0$$ Divide both sides by 24: $$x^2+2x+1=0$$ This can be factored to $(x+1{)}^2=0$. Thus, $x=-1$. Now, we need to test the intervals using the second derivative to determine the concavity and inflection point: 1. When $x<-1$: Choose a test value, say $x=-2$. Then $f^{″}(-2)=24(-2{)}^2+48(-2)+24=24>0$. Since the second derivative is positive here, the function is concave up in this interval. 2. When $x>-1$: Choose a test value, say $x=0$. Then $f^{″}(0)=24(0{)}^2+48(0)+24=24>0$. Since the second derivative is positive here, the function is also concave up in this interval. Since there is no change in concavity, there are no inflection points.

Final Answer:

The function, $$f(x)=2x^4+8x^3+12x^2-x-2$$, is concave up for all values of x (i.e., the whole domain) and there are no inflection points.

Key concepts

Second Derivative Test

The Second Derivative Test is a mathematical tool used to determine the concavity of a function and to identify potential inflection points. To perform this test, follow these simple steps:

1. **Find the Second Derivative**: Start by finding the second derivative of the function, denoted as $f^{″}(x)$. This involves differentiating the first derivative $f^{\prime }(x)$ once more.

2. **Set $f^{″}(x)$ to Zero**: Solve the equation $f^{″}(x)=0$ to identify critical points that may indicate a change in concavity. These points are potential inflection points.

3. **Test Intervals for Concavity**: Check the sign of $f^{″}(x)$ on intervals around these critical points. Choose test values in each interval to determine whether $f^{″}(x)$ is positive or negative.

• If $f^{″}(x)>0$ in an interval, the function is concave up on that interval.
• If $f^{″}(x)<0$ in an interval, the function is concave down on that interval.

In the context of the exercise, $f^{″}(x)$ was found to be positive over all intervals, confirming that the function is always concave up. Thus, there is no change in concavity and no inflection point.

Derivative

The derivative of a function provides us with the rate at which the function's value changes with respect to changes in its variable. It acts as an essential tool for analyzing and predicting the behavior of functions.

The first derivative $f^{\prime }(x)$ is derived from the original function and provides insights into the slopes or gradients along the curve of a function. It helps in identifying increasing or decreasing intervals.

Once you have the first derivative, you can proceed to find the second derivative, which is the derivative of $f^{\prime }(x)$. It offers additional information about the curve's concavity:

• If the second derivative $f^{″}(x)>0$, the curve is concave up (shaped like a "U").
• If the second derivative $f^{″}(x)<0$, the curve is concave down (shaped like an "n").

In the exercise, the first derivative was used to then calculate the second derivative, ultimately revealing the uninterrupted concavity of the function, as it remained positive.

Inflection Points

Inflection points are special kinds of points on the curve of a function where the concavity changes direction. Fortunately, the second derivative facilitates their identification.

To find inflection points, follow these steps:

1. Compute the Second Derivative: Start by ensuring you have the function's second derivative, $f^{″}(x)$.

2. Solve $f^{″}(x)=0$: This equation is solved to locate potential inflection points. These points might be where the concavity changes direction.

3. Assess Neighborhoods: Examine intervals around these identified points to affirm a change in the sign of $f^{″}(x)$. This will confirm the presence of an actual inflection point.

However, as demonstrated in the exercise, an inflection point occurs only if there is an actual change in concavity. Since the second derivative remained positive on all intervals for $f(x)$, it implies no inflection points exist, as there was no shift from concave up to concave down or vice versa.