SOLUTION: If x is a real number, find the minimum value of {{{ 3^(x^2 - 4x+8)}}} Please Help! Thanks to you all!

Algebra ->  Exponents -> SOLUTION: If x is a real number, find the minimum value of {{{ 3^(x^2 - 4x+8)}}} Please Help! Thanks to you all!      Log On


   

Question 831607: If x is a real number, find the minimum value of
+3%5E%28x%5E2+-+4x%2B8%29
Please Help! Thanks to you all!
Found 2 solutions by Alan3354, josmiceli:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
If x is a real number, find the minimum value of
+3%5E%28x%5E2+-+4x%2B8%29
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It's at the min value of x%5E2+-+4x%2B8
2x - 4 = 0
x = 2
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+3%5E%282%5E2+-+4%2A2+%2B+8%29+=+3%5E4
= 81

Answer by josmiceli(19441) About Me  (Show Source):
You can put this solution on YOUR website!
If I say
+f%28x%29+=+x%5E2+-+4x+%2B+8+
then +3%5E%28x%5E2+-+4x%2B8%29+ is smallest when
+f%28x%29+ is a minimum.
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When the form is +a%2Ax%5E2+%2B+b%2Ax+%2B+c+, then
the minimum is at +x%5Bmin%5D+=+-b%2F%282a%29+
+a+=+1+
+b+=+-4+
+-b%2F%282a%29+=+-%28-4%29%2F%282%2A1%29+
+-b%2F%282a%29+=+2+
+x%5Bmin%5D+=+2+
plug this back into the equation:
+f%282%29+=+2%5E2+-+4%2A2+%2B+8+
+f%282%29+=+4+-+8+%2B+8+
+f%282%29+=+4+
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Now I can say the minimum is at
+3%5E4+=+81+
Here's a plot of +3%5E%28x%5E2+-+4x%2B8%29+ and
+x%5E2+-+4x+%2B+8+
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