Question

(a) For a given function f(x) find i) those open intervals over which the function is concave up, concave down i) find all points of inflection. fx) = x² +x²2 +10​

Answer 1

Answer:

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Answer 2

Answer:

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Step-by-step explanation:

Example 1: Determine the concavity of f(x) = x 3 − 6 x 2 −12 x + 2 and identify any points of inflection of f(x).

Because f(x) is a polynomial function, its domain is all real numbers.

Testing the intervals to the left and right of x = 2 for f″(x) = 6 x −12, you find that

hence, f is concave downward on (−∞,2) and concave upward on (2,+ ∞), and function has a point of inflection at (2,−38)

Example 2: Determine the concavity of f(x) = sin x + cos x on [0,2π] and identify any points of inflection of f(x).

The domain of f(x) is restricted to the closed interval [0,2π].

Testing all intervals to the left and right of these values for f″(x) = −sin x − cos x, you find that

hence, f is concave downward on [0,3π/4] and [7π/4,2π] and concave upward on (3π/4,7π/4) and has points of inflection at (3π/4,0) and (7π/4,0).