Answer:
f(x) is concave up whenever:
B. 3x²−10 is positive
f(x) is concave down whenever:
A. 3x²−10 is negative
The points of inflection of f(x) are the same as:
B. the zeros of 3x²−10
Explanation:
Given the function f(x) = 1 / (x²+10)
We can determine the concavity by finding the second derivative.
If
f"(x) > 0 ⇒ f(x) is concave up
If
f"(x) < 0 ⇒ f(x) is concave down
Then
f'(x) = (1 / (x²+10))' = -2x / (x²+10)²
⇒ f"(x) = -2*(10-3x²) / (x²+10)³
if f"(x) = 0 ⇒ -2*(10-3x²) = 0 ⇒ 3x²-10 = 0
f(x) is concave up whenever 3x²−10 > 0
f(x) is concave down whenever 3x²−10 < 0
The points of inflection of f(x) are the same as the zeros of 3x²-10
it means that 3x²-10 = 0