Question

In Exercises, graph and analyze the function. Include extrema, points of inflection, and asymptotes in your analysis. $$ f(x)=x e^{-x} $$

Short answer

The function \(f(x) = xe^{-x}\) has a local maximum at \(x = 1\), a point of inflection at \(x = 0.5\), and a horizontal asymptote at \(y = 0\).

Step-by-step solution

Prime Derivative and Extrema

The derivative of \(f(x) = xe^{-x}\) can be found using the product rule:\n \[f'(x)=e^{-x}-xe^{-x}\]\n\nSetting \(f'(x) = 0\) to find the critical points:\n\[e^{-x}(1-x)=0\]\nThis gives us \(x = 1\).\n\nChecking the value before and after \(x=1\) in \(f'(x)\), we see that \(f'(x)\) is positive before \(x=1\) and negative after, which indicates that the function has a maximum at \(x = 1\).

Double Prime Derivative and Inflection Points

The second derivative of \(f(x)\) is handled similarly and can be calculated as:\n\[f''(x) = -e^{-x} + 2xe^{-x}\]\n\nSetting \(f''(x) = 0\) to find any inflection points:\n\n\[-e^{-x}(1-2x)=0\]\nThis gives \(x = 0.5\), which indeed changes the concavity of the function, so it's a point of inflection.

Asymptotes

There are no vertical asymptotes in the function.\n\nTo find horizontal asymptotes, investigate the behavior of \(f(x)\) as \(x\) approaches positive and negative infinity:\n\nAs \(x \rightarrow \infty\), \(f(x) \rightarrow 0\).\nAs \(x \rightarrow -\infty\), \(f(x) \rightarrow 0\).\n\nSo \(y = 0\) is a horizontal asymptote.

Graphing the function

Now gather the information and graph the function. The graph has a local maximum at \(x = 1\), a point of inflection at \(x = 0.5\), and a horizontal asymptote at \(y = 0\).

Key concepts

Extrema of Functions

When graphing exponential functions such as f(x) = xe^{-x}, identifying the extrema, or extreme values, is a crucial step. The extrema include the maximum and minimum values that the function can attain. To find these points, we first calculate the derivative of the function and set it equal to zero to determine the critical points.

As in our example, deriving f'(x) = e^{-x} - xe^{-x}, we find that the function has a critical point at x = 1. By analyzing the sign of f'(x) before and after x = 1, we conclude that f(x) has a local maximum at that point, since f'(x) changes from positive to negative. This is a piece of crucial information for sketching the graph and understanding behavior of the function.

Points of Inflection

Points of inflection are where the function changes concavity—in other words, it switches from curving upwards to downwards, or vice versa. To locate these points, the second derivative of the function, f''(x), is used.

For f''(x) = -e^{-x} + 2xe^{-x}, setting f''(x) = 0 leads to the point of inflection at x = 0.5. Around this value, the sign of f''(x) changes, indicating change in concavity of f(x). Inflection points are critical for drawing the curves of the graph accurately and understanding the function's geometry.

Asymptotes

Asymptotes are lines that the graph of a function approaches but never actually reaches. They can be vertical, horizontal, or oblique. In our example, there are no vertical asymptotes because the function f(x) = xe^{-x} is defined for all real numbers x.

For horizontal asymptotes, we examine the limits of f(x) as x approaches infinity or negative infinity. For both, our function approaches zero, so we can say that y = 0 is a horizontal asymptote. It's a vital feature as it gives us a boundary condition for the behavior of the graph at the extremes.

Product Rule Derivation

The product rule is a formula used to find the derivative of the product of two functions. The rule is given by (fg)' = f'g + fg', where f and g are functions of x. To derive it, we can begin with the limit definition of a derivative and apply it to the product of two functions.

For our example, the function f(x) = xe^{-x} is the product of x and e^{-x}. Applying the product rule, we get the first derivative f'(x) = e^{-x} - xe^{-x}. This rule helps us determine the behaviour of the function, including the critical points and ultimately the extrema.