Question 1106859: Identify intervals on which the function is increasing, decreasing, or constant.
g(x) = 1 - (x - 6)^2
Found 3 solutions by josgarithmetic, JThomson, greenestamps:
Answer by josgarithmetic(39625)   (Show Source):
You can put this solution on YOUR website! Vertex maximum
Vertex (6,1)
Increasing from negative infinity to 6;
Decreasing from 6 to infinity;
Maximum value for g when x is 6.
Answer by JThomson(12)  (Show Source):
You can put this solution on YOUR website! For the curve g(x) to increase, its gradient must be greater than 0 or must be positive;
g'(x) = -2(x-6), where g'(x) is the first derivative of g;
-2(x-6) > 0
x-6 < 0
x < 6
Answer by greenestamps(13203)  (Show Source):
You can put this solution on YOUR website!
The calculus approach used by one of the other tutors is fine. But not if you don't know calculus.
The solution by the other tutor is correct; but without explanation it might not be of much help.
The given function graphs as a parabola, because it (in its expanded form) has an x^2 term. In the given function 1-(x-6)^2, the value of the expression in parentheses is always 0 or positive. So in evaluating the function for a particular value of x, you will be subtracting 0 or a positive number from 1.
That means the maximum value of the function is 1. And that maximum value occurs when (x-6)^2 is 0, which is when x is 6.
So we have a downward opening parabola, with vertex (maximum value) at (6,1). That means the function is increasing for x<6 and decreasing for x>6.
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