Mathematics

Mathematics

Problem

Consider the function y=(3x2+5)e−xy=(3x^{2}+5)e^{-x} for x≥0x\ge 0. a. (i) Show that lim⁡x→∞(3x2+5)e−x=0\displaystyle \lim_{x\to\infty}(3x^{2}+5)e^{-x}=0. b. (ii) Hence write the equation of any horizontal asymptote of the graph of y=(3x2+5)e−xy=(3x^{2}+5)e^{-x}. c. (iii) Determine the least integer NN such that ∣(3x2+5)e−x∣<0.01|(3x^{2}+5)e^{-x}|<0.01 whenever x≥Nx\ge N. Provide full justification of your answer.
Intuitive explanation

The key here is product rule differentiation combined with recognizing that e−xe^{-x} decays while the polynomial grows. To find where the function increases or decreases, you'll take the derivative, factor it carefully, and determine where it's positive or negative. The interplay between the growing quadratic term and the exponential decay creates a turning point—that's where the interesting behavior lives.

Intuitive explanation

The key here is product rule differentiation combined with recognizing that e−xe^{-x} decays while the polynomial grows. To find where the function increases or decreases, you'll take the derivative, factor it carefully, and determine where it's positive or negative. The interplay between the growing quadratic term and the exponential decay creates a turning point—that's where the interesting behavior lives.

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