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08 Dec 2017, 21:42
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Is \(a^4+b^4\) less than 32?
(1) \(a + b = 4\)
(2) \(a^2 + b^2 = 8\)
(1) \(a + b = 4\)
(2) \(a^2 + b^2 = 8\)
08 Dec 2017, 23:35
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Is \(a^4+b^4\) less than 32?
You can solve this problem by testing values or use the following properties:
(1) \(a + b = 4\).
According to the first property above. \(a^4 + b^4\) will be minimized given that \(a + b = 4\), when \(a = b = 2\). In this case \(a^4 + b^4 = 2^4 + 2^4 = 32\). So, the minumum possible value of \(a^4 + b^4\) is 32 (not less than 32). Sufficient.
(2) \(a^2 + b^2 = 8\)
You can use the same property (which would be easier) or square and test the second property:
\(a^4 + 2a^2b^2 + b^4 = 64\)
\(a^4 + b^4 = 64 -2a^2b^2\)
According to the second property \(2a^2b^2\) is maximized given that \(a^2 + b^2 = 8\), when \(a^2 = b^2 = 4\). So, the maximum value of \(2a^2b^2\) is 32. This makes the minimum value of \(a^4 + b^4 = 64 -2a^2b^2\) equal to 64 - 32 = 32. So, the minumum possible value of \(a^4 + b^4\) is 32 (not less than 32). Sufficient.
Answer: D.
hope it's clear.
You can solve this problem by testing values or use the following properties:
For a given sum of two numbers, the sum of even powers of these numbers is minimized, when they are equal. Simply put if it's given that \(x + y = 10\), then \(x^{(even \ integer)} + y^{(even \ integer)}\) is minimum when \(x = y = 5\). So, the minimum value of \(x^2 + y^2\), given that \(x + y = 10\) is \(5^2 + 5^2 = 50\).
For a given sum of two numbers their product is maximized when they are equal. Simply put if it's given that \(x + y = 10\), then xy is maximized when \(x = y = 5\). So, the maximum value of xy, given that \(x + y = 10\) is 5*5 = 25.
For a given sum of two numbers their product is maximized when they are equal. Simply put if it's given that \(x + y = 10\), then xy is maximized when \(x = y = 5\). So, the maximum value of xy, given that \(x + y = 10\) is 5*5 = 25.
(1) \(a + b = 4\).
According to the first property above. \(a^4 + b^4\) will be minimized given that \(a + b = 4\), when \(a = b = 2\). In this case \(a^4 + b^4 = 2^4 + 2^4 = 32\). So, the minumum possible value of \(a^4 + b^4\) is 32 (not less than 32). Sufficient.
(2) \(a^2 + b^2 = 8\)
You can use the same property (which would be easier) or square and test the second property:
\(a^4 + 2a^2b^2 + b^4 = 64\)
\(a^4 + b^4 = 64 -2a^2b^2\)
According to the second property \(2a^2b^2\) is maximized given that \(a^2 + b^2 = 8\), when \(a^2 = b^2 = 4\). So, the maximum value of \(2a^2b^2\) is 32. This makes the minimum value of \(a^4 + b^4 = 64 -2a^2b^2\) equal to 64 - 32 = 32. So, the minumum possible value of \(a^4 + b^4\) is 32 (not less than 32). Sufficient.
Answer: D.
hope it's clear.
General Discussion
08 Dec 2017, 23:39
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diota2004
Is \(a^4+b^4\) less than 32?
(1) \(a + b = 4\)
(2) \(a^2 + b^2 = 8\)
(1) \(a + b = 4\)
(2) \(a^2 + b^2 = 8\)
(1) a + b = 4
We can try using some plugging in of numbers. First thing that comes to mind is 'what if a & b are each equal?'. So when we put a=b=2, we get a^4 + b^4 = 16+16=32.
Now if we give different values to them, eg, put a=3, b=1, we have a^4 + b^4 = 81+1 =82, or if we put a=0, b=4, we have a^4 + b^4 = 0+256 =256. Even if we give negative value to one of a or b, it wont matter because powers are even, it will convert to positive only. eg, if a=-1, b=5, we have a^4 + b^4 =1+625=626
So this plugging of numbers tells me that no matter what values to a&b I give, if a+b = 4, then a^4 + b^4 will always be greater than or equal to 32 (least value 32 will occur when a & b are equal). so a^4 + b^4 can never be less than 32. Sufficient
(2) a^2 + b^2 = 8
Note that a^4 = (a^2)^2 and b^4 = (b^2)^2. So we can proceed with same plugging in of numbers. First lets put a^2=b^2 = 4. In this case a^4 + b^4 = 4^2 + 4^2 = 16+16=32. Now if we substitute unequal values of a^2/b^2, eg, a^2 = 7, b^2 = 1, we get a^4 + b^4 = 49+1=50. Again, no matter which values we give to a^2 and b^2, as long as a^2+b^2 = 8, we will always have a^4 + b^4 as greater than or equal to 32 (least value 32 will occur when a^2 & b^2 are equal). So a^4 + b^4 can never be less than 32. Sufficient.
Hence D answer
