Question

the interval (s) over which the function f(x) is concave u(p)/(c)oncar the x-coordinates of the points of inflection for f(x) 2(x+1)(x-1)²(x-5)³=c

Answer 1

To identify where function f(x) = 2(x+1)(x-1)^2(x-5)^3 is concave up or down and the points of inflection, find the second derivative, set it to zero, and analyze sign changes in concavity around those points.

Step-by-step explanation:

To determine the interval(s) over which the function f(x) = 2(x+1)(x-1)²(x-5)³ is concave up or concave down and to identify the x-coordinates of the points of inflection, we need to find the second derivative of f(x) and analyze its sign changes. The points of inflection occur where the second derivative is zero or does not exist, provided that there is a sign change in the concavity.

First, take the first derivative f'(x). Then take the second derivative f''(x) by differentiating f'(x). After finding f''(x), set it equal to zero to find potential inflection points. Ensure that there's an actual change in the sign of f''(x) at these points to confirm them as inflection points. The function is concave up where f''(x) > 0 and concave down where f''(x) < 0.