True or False If `f: R->R` and `c in R` is such that f is increasing in `(c-delta, c)` and f is decreasing in `(c, c + delta)` then f has a local maximum at c. Where `delta` is a sufficiently small positive quantity.

Write True/False: If f'( c )=0 then f(x) has a local maximum or a local minimum at x=c

Write True/False: If f'( c ) exists and f'( c )gt0 , then f(x) has a local minimum at x=c.

True or False statements: If a function f is such that f' (c ) = 0, then at the point, c the function f has either a local maximum or a local minimum.

Lets have first derivative at c such that f'(c) = 0 and f'(c) gt 0, AA x in (c- delta c) , also f'(c) gt 0, AA x in (c,c+delta) , then c is a point of

Write True/False: If f'( c ) does not exist and f'(x) changes from positive to negative as x increases through c, then f(x) has a local minimum at x=c.

True or False statements: If a function f is twice differentiable at x = c,f' (c ) = 0 and f" (c ) > 0, then f has a local minimam value at x = c.

Let f : [a, b] rarr R be a function such that for c in (a, b), f'(c) = f''(c) = f'''(c)= f^(iv) (c) = f^(v)(c) = 0 . Then a)f has a local extremum at x = c b)f has neither local maximum nor minimum at x = c c)f is necessarily a constant function d)it is difficult to say whether (a) or (b).

Fill in the blanks: If a function f has a local maximum at x = c and f" (c ) < 0, then f' (c ) is equal to __________.

Write True/False: If f(x) has an absolute maximum at x=c, then f'( c )=0

In Delta ABC , if a = 10 and b cot B + c cot C = 2(r + R) then the maximum area of DeltaABC will be