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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblema 4.3.85
Capitolo 4, Problema 4.3.85

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


f(x) = x²e⁻ˣ

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First, find the first derivative of the function \( f(x) = x^2 e^{-x} \). Use the product rule, which states that if you have a function \( u(x)v(x) \), its derivative is \( u'(x)v(x) + u(x)v'(x) \). Here, let \( u(x) = x^2 \) and \( v(x) = e^{-x} \).
Calculate the derivatives: \( u'(x) = 2x \) and \( v'(x) = -e^{-x} \). Substitute these into the product rule to find \( f'(x) = 2x e^{-x} + x^2 (-e^{-x}) \). Simplify to get \( f'(x) = e^{-x} (2x - x^2) \).
Set the first derivative \( f'(x) = e^{-x} (2x - x^2) \) equal to zero to find the critical points. Since \( e^{-x} \) is never zero, solve \( 2x - x^2 = 0 \). Factor the equation to get \( x(2 - x) = 0 \), giving critical points at \( x = 0 \) and \( x = 2 \).
Next, find the second derivative \( f''(x) \) to apply the Second Derivative Test. Differentiate \( f'(x) = e^{-x} (2x - x^2) \) using the product rule again. Let \( u(x) = e^{-x} \) and \( v(x) = 2x - x^2 \).
Calculate \( f''(x) = e^{-x}(-1)(2x - x^2) + e^{-x}(2 - 2x) \). Simplify this expression. Evaluate \( f''(x) \) at the critical points \( x = 0 \) and \( x = 2 \) to determine the nature of these points: if \( f''(x) > 0 \), it's a local minimum; if \( f''(x) < 0 \), it's a local maximum.

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Critical Points

Critical points of a function occur where its first derivative is zero or undefined. These points are essential for identifying potential local maxima and minima, as they represent locations where the function's slope changes. To find critical points, one must differentiate the function and solve for the values of x that satisfy the condition f'(x) = 0.
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Critical Points

Second Derivative Test

The Second Derivative Test is a method used to classify critical points as local maxima, local minima, or saddle points. It involves evaluating the second derivative of the function at the critical points. If f''(x) > 0, the point is a local minimum; if f''(x) < 0, it is a local maximum; and if f''(x) = 0, the test is inconclusive.
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The Second Derivative Test: Finding Local Extrema

Exponential Functions

Exponential functions, such as f(x) = x²e⁻ˣ, combine polynomial and exponential components. The behavior of these functions is influenced by both the polynomial growth and the decay of the exponential part. Understanding their derivatives is crucial for applying the critical point analysis and the Second Derivative Test effectively.
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Exponential Functions
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