The given function is expressed as
f(x) = 2.7 + 4.6x - 1.8x^2
a) To find the critical numbers, the first step is to differentiate the function.
Recall, if f(x) = ax^b,
By differentiating it, it becomes
f'(x) = abx^(b - 1)
If we differentiate a constant, the result is 0
Thus, by differentiating the given function, it becomes
f'(x) = 4.6 - 1.8 * 2x
f'(x) = 4.6 - 3.6x
To find the critical number, we would equate the derivative to 0 and solve for x. It becomes
4.6 - 3.6x = 0
3.6x = 4.6
x = 4.6/3.6
x = 1.3
The critical number is x = 1.3
b) To find the intervals where the function is increasing or decreasing, we would test the values of x on the left and right of the critical number.
On the left, we would test x = 1.2
It becomes
f'(x) = 4.6 - 3.6 * 1.2 = 4.6 - 4.32 = 0.28
Since it is greater than zero, it means that f(x) is increasing on this interval. The interval where it is increasing is (- infinity, 1.3)
c) On the right, we would test x = 1.4
It becomes
f'(x) = 4.6 - 3.6 * 1.4 = 4.6 - 5.04 = - 0.44
Since it is less than zero, it means that f(x) is decreasing on this interval. The interval where it is decreasing is (1.3, infinity)