Suppose f:R rarr R has two continuous derivatives, has only one critical point x_0, and that f′′(x_0)<0. Then f achieves its global maximum at x_0, that is f(x)≤f(x_0) for all x in R.

Taniya Burns

Taniya Burns

Answered question

2022-07-22

Suppose f : R R has two continuous derivatives, has only one critical point x 0 , and that f ( x 0 ) < 0. Then f achieves its global maximum at x 0 , that is f ( x ) f ( x 0 ) for all x R .

Answer & Explanation

sweetwisdomgw

sweetwisdomgw

Beginner2022-07-23Added 20 answers

The function f is continuous. It has a unique 0 at x 0 . Since f ( x 0 ) < 0 the function f is decreasing around x 0 . So just below x 0 you have f is positive and just above x 0 it is negative.
Now, as f is continuous and by the intermediate vlaue theorem you get that f is positive for all x < x 0 and f is negative for all x < x 0 . So, f is increasing for x < x 0 and decreasing for x < x 0 , which yields the claim.

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