Suppose f : R → R has two continuous derivatives, has only one critical point x 0 , and that f ″ ( x 0 ) &lt; 0. Then f achieves its global maximum at x 0 , that is f ( x ) ≤ f ( x 0 ) for all x ∈ R .

Question

Suppose   f  :      R    →      R   has two continuous derivatives, has only one critical point       x    0  , and that       f    ″    (      x    0    )  &amp;lt;  0. Then   f achieves its global maximum at       x    0  , that is   f  (  x  )  ≤  f  (      x    0    ) for all   x  ∈      R  .

sweetwisdomgw · Accepted Answer

The function       f    ′   is continuous. It has a unique   0 at       x    0  . Since       f    ″    (      x    0    )  &amp;lt;  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  &amp;lt;      x    0   and       f    ′   is negative for all   x  &amp;lt;      x    0  . So, f is increasing for   x  &amp;lt;      x    0   and decreasing for   x  &amp;lt;      x    0  , which yields the claim.

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.

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