Determine the largest open interval of the domain (a) over which ...

Question

Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. See Example 2. ƒ(x) = x^2 - 4x + 3

Accepted Answer

Consider the function F of X is equals to X squared plus 12 X plus and find the largest open interval of the domain at which the function is increasing or decreasing. Our possible answers are some different interval options. So we have F of X as equals to X squared plus 12 x plus 40. To solve this, we want to change our quadratic first to standard form of a Parabola. Standard form of Parabola is given by F of X is equals to X minus H squared plus K. We have X squared plus 12 X plus 40. Now to do this, we will complete the square playing the squares means that we will take the B and our quadratic 12th, divide it by two and square. It, we'll add it to both sides. We have X squared plus 12 X plus plus 40 as a result. Now, because we added 36 and we want the other form we have, we'll subtract 36 as well. 36 -36. We go to zero, which would give us our original form. We can then rewrite our X squared plus 12 X plus as a factor X plus six squared, 40 -36 will be plus four. So we have X plus six squared plus four. We can now determine our vertex H K to be -64. Like so now that we have our vertex, we wanna check values to the left and to the right of a vertex point. So a vertex point happens at X equals negative six. Let's make a little table and check two values a check of value less than 86. We will say -7. Also check the value greater than -6. You'll say -5. No, let me find what our F -7 is. That would be negative seven squared plus 12 multiplied by negative seven plus 40 which simplifies to five but negative five, that'd be F of negative seven S equals to negative five squared plus 12 multiplied by negative five plus 40 which is also five. Now we notice to the right, our eyes are increasing and to the left, our Y vis are also increasing. We can then determine that we have an upwards facing parabola on upwards facing Parabola. Anything to the left of the vertex will be decreasing to the right will be increasing. Which means our increasing interval will be from native affinity. I'm sorry. That's our decreasing interval. Our increasing interval will be from our vertex -6 to infinity. Our decreasing interval will be from negative affinity to negative six Yeah, we can then determine the answer to our problem will be answer. C ok. I hope to help you solve the problem. Thank you for watching. Goodbye.

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Determine the largest open interval of the domain (a) over which the function is increasing and (b) over which it is decreasing. See Example 2. ƒ(x) = x^2 - 4x + 3

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