Find the function blanks

Question

Let

f′(x)=4−3x2�′(�)=4−3�2

be the derivative of a continuous function f�. Fill in the blanks below.

1.	f� is increasing for x∈�∈	functions
2.	f� is decreasing for x∈�∈	functions
3.	The local maxima of f� occur at x=�=	functions
4.	The local minima of f� occur at x=�=	functions
5.	f� is concave upward for x∈�∈	functions
6.	f� is concave downward for x∈�∈	functions
7.	The x�-values of the point(s) of inflection of f� occur(s) at x=�=	functions

Answer 1

NOTE: I found a similar problem online, so I'm guessing the derivative is f'(x) = 4 - 3x². If that's not correct please let me know.

First, let's find where that derivative is equal to zero. This would occur when x = ±²/_√3. If x is between those values, the derivative is positive. If x is not between the two values, the derivative is negative. This means for 1) and 2), the function is increasing on the interval (-²/_√3, ²/_√3) and decreasing (-∞, ²/_√3)∪(²/_√3, ∞).

Next, let's find the second derivative since we need it for 5-7. f''(x) = -6x. This would change from negative for x<0 to positive for x>0. So,

5) The function is concave up for x in the interval (-∞, 0). This also means that the local minima for 4) is at x = -²/_√3 because the graph is concave up at that location.

6) The function is concave down for x in the interval (0, ∞). This also means that the local maxima for 3) is at x = ²/_√3 because the graph is concave down at that location.

7) Because the concavity changes from positive to negative at x=0, this is our inflection point.