Question

True or False Evaluate each of the statements below as true or false. 

a) A critical number of a function f(x) can only be defined as a number c in the domain of f(x) where f'(c) = 0). 

b) If f(x) is a continuous function on [a,b], then an absolute maximum ſ(c) and an absolute minimum f(d) for some c, d in [a, b] are guaranteed to exist. 

c) If f'(c) = 0, then f has a local maximum or a local minimum at x = c

d) If f'(x) < 0 for 1 < x < 6, then f is decreasing on (1,6).

Answer 1

ANSWER :

a.  TRUE 

Critical point exists when tangent at that point is horizontal I.e. slope = f’(c) = 0.

b. FALSE.

Maxima and minima, both, may exist or may not exist at two different points in the interval. It can not be guaranteed.

c. TRUE

If f’(c) = 0, then a local maxima or a local minima or both (inflection point) will exist.

d. TRUE

If f’(x) is negative in some interval, then function will be decreasing in that interval.

Answer 2

Sol:- is defined as 6 True Because Cantica Point thei Value of c for for which f'co=0 Taus Because fro) is Continous an [a, b] and fix) is limited toa: frond limot gueranteed to have Tanimum Ainimum Valve. ہے۔ رة ملے 0 FALSE f'(C) = 0 (& some be a Then Point times e might (en saddle Point. of Saddle CSP) TRUE "Because f'ww co end then frac is decreasing an I <3<6 (1,6).