The problem is: \" Let P = { f(x) = a + bx + cx^2 | a,b,c are real numbers} . Th
ID: 2968377 • 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]
F = { f(x) is a member of P | f(0) = f(1) = f(2) = 0}
G = { f(x) is a member of P | f "(0) = f '(0) = 0} [Note:The value of the second and first derivatives at 0 are 0].
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 F
ii. F is a subset of C.
iii. C is a subset of G
iv. G 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
C = { f(x) is a member of P | f '(0) = 0}
f(x) = a + bx + cx^2
f'(x) = b + 2cx
f'(0)= b = 0
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F = { f(x) is a member of P | f(0) = f(1) = f(2) = 0} ----(1)
f(0)= a, f(1)= a + b +c , f(2) = a + 2b + 4c
f(0) = a = 0 from (1)----(2)
f(1) = a + b + c = 0+b+c =0 => b = -c
f(2) = a + 2b + 4c = 0-2c + 4c = 2c = 0 => c = 0
which means a= b= c = 0.
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for subset C, b=0 , a and c can take any real numbers including 0.
for subset F, a,b, and c all has 0 value. So F is a special case of C i.e. F is a subset of C------answer for(ii)
but C is not a subset of F
example assume a=1 , b=0 and c=1
f'(0) = 0 which satiesfies condition of C
but f(0)= 1, f(1)=2 , and f(2)= 5 i.e. none of them are 0.--------answer for (i)
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G = { f(x) is a member of P | f "(0) = f '(0) = 0}
f'(x) = b + 2cx
f'(0)= b = 0.
f''(x)= 2c
f''(0).= 2c=0
c=0
for subset C, b=0 , a and c can take any real numbers including 0.
for subset G, b=c=0 and a can take any real numbers including 0.. So G is a subset of C------answer for(iv)
but C is not a subset of G
example assume a=1 , b=0 and c=1
f'(0) = 0 which satiesfies condition of C
but f''(0)= 2 --------answer for (iii)
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