Question

Sketch the graph of a single function f that satisfies all of the conditions below.

(a) f '(x) < 0 on (1,∞), f '(x) > 0 on (−∞,1)

(b) f ''(x) > 0 on (−∞,−2) and (2,∞), f ''(x) < 0 on (−2,2)

(c) lim x→−∞ f(x) = −2, lim x→∞ f(x) = 0

Answer 1

we have to use following condition to determine given data is sufficient or not

* when f'(x) >0 => function increasing graph y=f(x) is moving upward

* when f'(x) <0 => function decreasing graph of y=f(x) moving downward

* at critical point f '(x) =0 or undefined , function changes from increasing to decreasing or vice versa at critical point   

* If function changes from increasing to decreasing at critical point the that critical point become point of maximum [ f"(x) <0 ]

* If function changes from decreasing to increasing at critical point the that critical point become point of minimum [f "(x) >0 ]

*  If f(x) is concave downward ( i.e. curve lies below all  tangents in that intervals ) then 2nd derivative f "(x) become negative [ f "(x)<0 ]

* If f(x) is concave upward ( i.e. curve lies above  all  tangents in that intervals ) then 2nd derivative f "(x) become positive   [ f "(x)>0 ] .

* if curve changes its curvature i.e. changes from concave upward to concave down ward or vice then that point is called as   point of inflection at point of inflection f”© =0

* Also we have to check for horizontal assymptotes

for horizontal asymptotes are horizontal line ( parallel to x-axis ) which will touch the curve at infinity )

i.e. y= a will be horizontal asymptotes if x tending to infinity (+,-) gives value of function f(x) equal to a [ a must be finite vale ]

if then y=a is horizontal asymptote