Concavity: The function f(x) = (x^3+3x^2+3x+1)/(x^2+2x-35)
is defined on the interval [-20,19]. The vertical asymtotes are x=5, -7. Then, f(x) is concave down on the region_____?

Question

Concavity: The function f(x) =  (x^3+3x^2+3x+1)/(x^2+2x-35)
is defined on the interval [-20,19]. The vertical asymtotes are x=5, -7. Then, f(x) is concave down on the region_____?

anonymous · Answer

Take the second derivative and find out where it changes sign (set it equal to zero and find solutions).  The curve is concave down where the second derivative is negative, which means the slope (first derivative) is decreasing, and therefore the curve is increasing at an ever decreasing rate, then going negative, causing it to "curve down"