Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

5. Graphical Applications of Derivatives

Finding Global Extrema

Calculus

5. Graphical Applications of Derivatives

Finding Global Extrema

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Multiple Choice

For which values of $x$ is the function $f (x) = (x^{2} - 3) e^{- x}$ increasing?

x = 1

x < 1

x < - 1

x > 1

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Verified step by step guidance

Step 1: To determine where the function f(x) = (x^2 - 3)e^{-x} is increasing, calculate its derivative f'(x). Use the product rule since f(x) is the product of two functions: u(x) = x^2 - 3 and v(x) = e^{-x}. The product rule states: (uv)' = u'v + uv'.

Step 2: Compute the derivative of u(x) = x^2 - 3, which is u'(x) = 2x. Compute the derivative of v(x) = e^{-x}, which is v'(x) = -e^{-x}. Substitute these into the product rule formula to find f'(x).

Step 3: Combine the results to express f'(x): f'(x) = (2x)e^{-x} + (x^2 - 3)(-e^{-x}). Simplify the expression: f'(x) = e^{-x}(2x - x^2 + 3).

Step 4: Factorize the simplified derivative f'(x) = e^{-x}(3 - x^2 + 2x). Since e^{-x} is always positive for all x, the sign of f'(x) depends only on the quadratic expression (3 - x^2 + 2x). Rewrite it as: -x^2 + 2x + 3 = -(x^2 - 2x - 3). Factorize the quadratic: -(x - 3)(x + 1).

Step 5: Analyze the sign of -(x - 3)(x + 1). The function f(x) is increasing where f'(x) > 0. Solve the inequality -(x - 3)(x + 1) > 0 by testing intervals determined by the roots x = -1 and x = 3. Determine the intervals where the product is positive and conclude the values of x where f(x) is increasing.