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) = -3x^2 + 18x + 1

Accepted Answer

Consider the function F of X is equals to negative two X squared plus 12 X minus 23 find the largest open interval of domain at which the function is increasing and decreasing. And so we have our possible answers being increasing and decreasing intervals. You wanna find what interval we're having? Increasing and decreasing values. So to do that, we wanna check the vertex of our parabola refers to the general form of a problem which is given by X minus H squared plus K like so we'll take our vertex as H comma K need to change our F FX into this form. We can start by factoring out negative two from our X squared and 12 X negative two multiplied by X squared minus six X minus 23. Now, to get into that form, we use to complete the square to complete the square. We take R B coefficient and our para divide it by two square. It add it both inside parentheses and outside the parentheses. So we will get negative two multiplied by X squared minus six X plus negative six, divided by two squared minus 23. Then we add our completed square, -6 divided by two square. Now, one other thing to note is that because we have a two multiplier on our entire parabola. We want to multiply this also by two, you will then get negative two multiplied by X squared minus six X plus nine minus plus 18. We can then simplify as negative two multiplied by X -3 Squared which tells us our vertex H K is at 3 -5. So we have our vertex, we wanna test the values on our equation. Let's make it a little table X Y. So we have 3 -5. We wanna test an X value that is less than our vertex and more than our vertex X value. Let's check 2 and 4 F of two will be negative two multiplied by two squared plus 12, multiplied by 2 - can simplify this. When it's simplified, we get negative seven, do the same thing for four. You would also get -7. Now we notice here Our Vertex on both sides of our vertex. Our graph is going down. In other words, we have a downward facing Parabola, something like that. Because you have a downward facing parabola are increasing will be from negative infinity to our vertex. Decreasing will be from our vertex to positive infinity. Which machines are increasing. You will say it's from negative infinity to our X corner of our vertex. three decreasing will then be from 3 to positive infinity. We can then conclude the answer to our problem will be answered deep. OK? I hope to help you solve the problem. Thank you for watching. Goodbye.

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) = -3x^2 + 18x + 1

Key Concepts

Derivative and Increasing/Decreasing Functions

Critical Points

Interval Notation

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