Suppose that f(x) = (x2 + 12) (1 - x2). Find all critical numbers of f. If there

Question

Suppose that f(x) = (x2 + 12) (1 - x2). Find all critical numbers of f. If there are no critical numbers, enter 'NONE'. Critical numbers = Use interval notation to indicate where f (x) is increasing. Note: Use 'INF' for infinity, '-INF' for -infinity, and use 'U' for the union symbol. Increasing: Use interval notation to indicate where f (x) is decreasing. Decreasing: Find the x-coordinates of all local maxima of f. If there are no local maxima, enter 'NONE'. X values of local maxima = Find the x-coordinates of all local minima of f. Note: If there are no local minima, enter 'NONE'. X values of local minima = Use interval notation to indicate where f (x) is concave up. Concave up: Use interval notation to indicate where f (x) is concave down. Concave down: List the x values of all inflection points of f. If there are no inflection points, enter 'NONE'. X values of inflection points = Find all horizontal asymptotes of f. If there are no horizontal asymptotes, enter 'NONE'. Horizontal asymptotes y = Find all vertical asymptotes of f. If there are no vertical asymptotes, enter 'NONE'. vertical asymptotes x = Use all of the preceding information to sketch a graph of f. When you're finished, enter a "1" in the box below. You may need to turn in your sketch in class. Graph Complete:

Explanation / Answer

f(x) = (x^2 +12) (1-x^2)

f'(x) = 0

2x (1-x^2) + (x^2 +12) *-2x = 0

=> 2x{1-x^2 -x^2 -12} = 0

=> x= 0 and -2x^2 = 11 => x = sqrt(11/2)

to find inflection point ,f"(x) = 0

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