Over what interval is each function increasing and over what interval is each function decreasing? (Table in the description )

Question

x f(x)=-0.3x^2 (x,y)

____|______________|________

-2 -1.2 (-2,-1.2)

-1 -0.3 (-1,-0.3)

0 0 (0,0)

1 -0.3 (1,0.3)

2 -1.2 (2,-1.2)

Answer 1

One method is to find the critical points of the function by setting its derivative equal to 0:

f(x) = -0.3x²

f'(x) = -0.6x

0 = -0.6x

x = 0

Next, we can use the test points x=-1 and x=1 to observe the behavior of the derivative around x=0:

f'(-1) = -0.3(-1)² = -0.3 < 0

f'(1) = -0.3(1)² = -0.3 < 0

As the derivative appears to always be negative, we can say the function is decreasing over the open interval (-∞,∞). Also, because the second derivative is f''(x)=-0.6 which is less than 0, this also leads us to the same conclusion that the function is concave down / always decreasing.

Hope this helps!