Get the solution to "How do you determine whether the function f(x)=−x4−9x3+2x+4..."

Christopher Stone

Answered question

2022-04-16

How do you determine whether the function

f (x) = - x^{4} - 9 x^{3} + 2 x + 4

is concave up or concave down and its intervals?

firenzesunzc65

Beginner2022-04-17Added 16 answers

Step 1
To find where a function is concave up, find where the second derivative of the function is positive.

f (x) = - x^{4} - 9 x^{3} + 2 x + 4

Find f'(x):

f^{'} (x) = - 4 x^{3} - 27 x^{2} + 2

Next, find f"(x):

f^{″} (x) = - 12 x^{2} - 54 x

f^{″} (x) = (- 6 x) (2 x + 9)

Set f"(x) equal to zero to find inflection points

0 = (- 6 x) (2 x + 9)

x = 0, x = - 4.5

Step 2
After checking the signs of values around these numbers, we find that f"(x) is positive on (-4.5, 0) i.e. convex and f"(x) is negative on

(- \infty, - 4.5) \cup (0, \infty)

i.e. concave.

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