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Find the maximum or minimum value of the function 𝑓(𝑥) = 𝑥² + 10, given 𝑥 ∈ [−3, 3].
Find the given maximum or minimum
value of the function 𝑓 of 𝑥 equals 𝑥 squared plus 10 given that 𝑥 is contained
in negative three, three.
First, let’s decide if we’re
dealing with the maximum or minimum. We’re given the function 𝑓 of 𝑥
equals 𝑥 squared plus 10. Because we have a positive 𝑥
squared value, the parabola will open upwards. And that means we’re dealing with a
minimum value. The minimum value is found at this
vertex point that we label ℎ, 𝑘.
To find the ℎ-value, we solve for
negative 𝑏 over two 𝑎 when quadratic equations are in the form 𝑎𝑥 squared plus
𝑏𝑥 plus 𝑐. We have 𝑥 squared plus 10, meaning
the 𝑏-value is zero and the 𝑎-value is one.
Plugging this in looks like this: ℎ
equals negative zero over two times one. Zero divided by anything is
zero. ℎ equals zero. The 𝑘-value will be equal to the
function of ℎ. Our ℎ-value is zero. And that means we’ll solve for zero
squared plus 10. 𝑘 equals 10.
Zero, 10 is the vertex, and it is
the minimum. And that means the minimum value is
10. The minimum outcome of our function
𝑓 of 𝑥 equals 𝑥 squared plus 10 has to be 10.
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