京都大学 情報学研究科 知能情報学専攻 2025年8月実施 情報学基礎 F1-2

Author

itsuitsuki

Description (English)

In the questions below, is the set of all real numbers and denotes Napier's constant (the base of the natural logarithm function).

Q.1

Consider the real-valued function defined on . Derive the maximum and minimum values of subject to the constraint that . Derivations must be clearly shown.

Q.2

Answer the following questions about the real-valued functions and . Let be a non-zero real number and be an integer greater than . Here, and denote the -th derivatives of and , respectively.

(1) Derive and .

(2) Derive .

(3) Derive .

Q.3

Answer the following questions. Derivations must be clearly shown.

(1) Find the following limit, if it exists. If the limit does not exist, prove it.

(2) Determine the set of for which the following series of functions converges.