Question

Let f(x)=5 x³ -2 x+6

(a) f is concave upward for x
(b) f is concave downward for x
(c) List the x-values of the points of inflection of f x=

Answer 1

The function f(x) = 5x³ - 2x + 6 is concave upward and the points of inflection are x = ±√(1/15).

Step-by-step explanation:

The function f(x) = 5x³ - 2x + 6 represents a cubic function. To determine whether the function is concave upward or concave downward, we need to analyze the second derivative. The second derivative of f(x) is f''(x) = 30x² - 2.

(a) If f''(x) > 0, then f(x) is concave upward.

(b) If f''(x) < 0, then f(x) is concave downward.

To find the points of inflection, we need to solve the equation f''(x) = 0. 30x² - 2 = 0. Solving this quadratic equation, we find x = ±√(2/30) = ±√(1/15).