very urgent Q1: The function f:a to b defined by f(x) = - x2+6x - 8 is a bijection - Maths - Limits and Derivatives

Question

very urgent

Q1: The function f:a to b defined by f(x) = - x2 +6x - 8
is a bijection if a and b belongs to which interval

Q2: If f(4) = 4 , f*(4) = 1 then lim x tends to 4
2-​​√f(x) / 2 -​√x is equal to

Ajanta Trivedi · Accepted Answer

the graph of the given function is as
follows:
<img alt="" src=
"https://s3mn%20mnimgs%20com/img/shared/ck-files/ck_55237eedefbf1%20png"
style="width: 650px; height: 400px;">
if we draw a line from the vertex parallel to y-axis, the given
curve is divided into two parts
one is (-∞, 3) other is (3,∞)in the first interval function is increasing and in the 2nd interval function is decreasing
and the range of the function is (-∞,1)therefore f:(-∞, 3)→(-∞,1) is a bijectionand f:(3,∞)→(-∞,1) is also a bijection

(2):
given: f(4)=4 and f'(4)=1limx→42-f(x)2-x , which is of the form 00therefore applying L'HOSPITAL rule:limx→42-f(x)2-x=limx→4-12f(x)
f'(x)-12x=limx→4f'(x)f(x) x
=f'(4)f(4) 4=14*2=12*2=14

hope this helps you

very urgent Q1: The function f:a to b defined by f(x) = - x2 +6x - 8 is a bijection if a and b belongs to which interval Q2: If f(4) = 4 , f*(4) = 1 then lim x tends to 4 2-​​√f(x) / 2 -​√x is equal to

very urgent

Q1: The function f:a to b defined by f(x) = - x² +6x - 8 is a bijection if a and b belongs to which interval

Q2: If f(4) = 4 , f*(4) = 1 then lim x tends to 4 2-√f(x) / 2 -√x is equal to