Test f(x)=−5x^4−6x^3+6 for concavity and inflection points?

To find inflection points and concavity, you must find the second derivative. So:

f'(x)= -20x^3-18x^2

f''(x)= -60x^2-36x

Now, to find inflection points, set the second derivative equal to 0 and solve for x.

-60x^2-36x = 0

-12x(5x+3) = 0

-12x = 0 and 5x+3=0

x = 0 and x= -3/5

Plug those x-values into the original function to get the y-values. You'll get (0 , 6) and (-3/5 , 831/125).

Now, figure out where the second derivative is negative (when the function is concave down) and positive (when the function is concave up). Let's use -1, -1/2, and 1.

f''(-1) = -24

f''(-1/2) = 3

f''(1) = -96

So, f''(-1) and f''(1) are negative, meaning the graph of the original function is concave down on the open intervals (-∞, -3/5) and (0, ∞), while f''(-1/2) is positive, meaning the graph of the original function is concave up on the open interval (-3/5, 0).