The problem is: \" Let P = { f(x) = a + bx + cx^2 | a,b,c are real numbers} . Th
ID: 2968372 • Letter: T
Question
The problem is:
" Let P = { f(x) = a + bx + cx^2 | a,b,c are real numbers} .
Then consider the following subsets of P:
C = { f(x) is a member of P | f '(0) = 0} [Note: the derivative of f(x) at 0 is equal to 0]
D = { f(x) is a member of P | f(0) = 0} [Note: the value of f(x) at 0 is equal to 0]
E = { f(x) = a | a is a real number} [Note: f(x) = a such that a is a real number]
For each of the containments below, state whether it is true or false. If true give a proof, if false, give an example of an element in the first set which is not contained in the second set:
i. C is a subset of D
ii. D is a subset of C.
iii. C is a subset of E.
iv. E is a subset of C. "
I am really stumped by this problem. I would greatly appreciate anyone's help, and I'll be sure to quickly give the points to the best answer. Thanks!
Explanation / Answer
1) False
C is not a subset of D
as
consider f(x) = 1+3x^2
clearly f(x) belongs to C but
f(0) =1 not equal to 0 hence f(x) doesn't belong to D
hence C is not a subset of D
2)False
D is not a subset of C
as consider
f(x) = 2x
clearly f(0) = 0
=> f(x) belongs to D
but f'(0) =2 not equal to 0
hence f(x) doesn't belong to C
hence
D is not a subset of C
3) False
E contains all the real valued constant functions
consider
f(x) = 2x^2
clear f'(x) =4x
=> f'(0) = 0
but f(x) is not a constant function
ie f(x) is not equal to a for all x
Hence C is not a subset of E
4) True
E contains all the real valued constant functions
Now
if some f(x) belongs to E
=> f(x) = a for all x for some a ;
=> f(x) is a constant function
=> f'(x) =0 for all x
=> f'(0) = 0
=> f(x) belongs to C
hence we proved if any f(x) belongs to E , it belongs to C too
hence E is a subset of C
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