Video: Finding the Maximum or Minimum Value of a Quadratic Function

Find the maximum or minimum value of the function 𝑓(π‘₯) = π‘₯Β² + 10, given π‘₯ ∈ [βˆ’3, 3].

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Video Transcript

Find the given maximum or minimum value of the function 𝑓 of π‘₯ equals π‘₯ squared plus 10 given that π‘₯ 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 π‘₯ equals π‘₯ squared plus 10. Because we have a positive π‘₯ 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 β„Ž, π‘˜.

To find the β„Ž-value, we solve for negative 𝑏 over two π‘Ž when quadratic equations are in the form π‘Žπ‘₯ squared plus 𝑏π‘₯ plus 𝑐. We have π‘₯ squared plus 10, meaning the 𝑏-value is zero and the π‘Ž-value is one.

Plugging this in looks like this: β„Ž equals negative zero over two times one. Zero divided by anything is zero. β„Ž equals zero. The π‘˜-value will be equal to the function of β„Ž. Our β„Ž-value is zero. And that means we’ll solve for zero squared plus 10. π‘˜ 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 π‘₯ equals π‘₯ squared plus 10 has to be 10.

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