To solve this problem, we need to evaluate the function [tex]\( f(x) = x^4 - 4x^3 + 10 \)[/tex] at the points in the partition [tex]\( P = \{1, 2, 3, 4\} \)[/tex]. Afterward, we will determine the upper sum [tex]\( U(P, f) \)[/tex] and the lower sum [tex]\( L(P, f) \)[/tex] for the function over this interval.
Step 1: Evaluate the function at the points in [tex]\( P \)[/tex].
For [tex]\( x = 1 \)[/tex]:
[tex]\[ f(1) = 1^4 - 4 \cdot 1^3 + 10 = 1 - 4 + 10 = 7 \][/tex]
For [tex]\( x = 2 \)[/tex]:
[tex]\[ f(2) = 2^4 - 4 \cdot 2^3 + 10 = 16 - 32 + 10 = -6 \][/tex]
For [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = 3^4 - 4 \cdot 3^3 + 10 = 81 - 108 + 10 = -17 \][/tex]
For [tex]\( x = 4 \)[/tex]:
[tex]\[ f(4) = 4^4 - 4 \cdot 4^3 + 10 = 256 - 256 + 10 = 10 \][/tex]
So, the function values at the points in [tex]\( P \)[/tex] are:
[tex]\[ f_P = \{ f(1), f(2), f(3), f(4) \} = \{ 7, -6, -17, 10 \} \][/tex]
Step 2: Find the upper sum [tex]\( U(P, f) \)[/tex].
The upper sum [tex]\( U(P, f) \)[/tex] is the maximum value of [tex]\( f(x) \)[/tex] over the partition points in [tex]\( P \)[/tex]:
[tex]\[ U(P, f) = \max\{ 7, -6, -17, 10 \} = 10 \][/tex]
Step 3: Find the lower sum [tex]\( L(P, f) \)[/tex].
The lower sum [tex]\( L(P, f) \)[/tex] is the minimum value of [tex]\( f(x) \)[/tex] over the partition points in [tex]\( P \)[/tex]:
[tex]\[ L(P, f) = \min\{ 7, -6, -17, 10 \} = -17 \][/tex]
Comparing these results with the given options, we find:
(i) [tex]\( U(P, f) = -17 \)[/tex] is incorrect.
(ii) [tex]\( L(P, f) = 11 \)[/tex] is incorrect.
(iii) [tex]\( U(P, f) = 40 \)[/tex] is incorrect.
(iv) [tex]\( L(P, f) = -40 \)[/tex] is incorrect.
Therefore, the correct values are:
[tex]\[ \boxed{\text{Neither of the given options is correct. The correct values are: } U(P, f) = 10 \text{ and } L(P, f) = -17.} \][/tex]
