Application of derivatives
I need help in Few questions in which I got stuck ;-(
Multiple answers(One or more correct answers) questions..
1)If ax+by+c=0 is normal to the curve y=x/(1+|x|), then a)ab>0 true
My problem : mod x is creating problem..Do I need to form 2 functions(for x>0 and for x<0) and then solve..
This is an odd function so it is enough to study x>0
2)The normal to the curve represented by parametrically by x=a(cosӨ+ӨsinӨ) and y=a(sinӨ-ӨcosӨ)at any point Ө such that it
a)makes a constant angle with x-axis
b)is at a connstant distant from the origin true
.c)touches a fixed circle. true
d)passes through the origin.
My solution: I'm gettin the eqn. of the normal as
y-a(sinӨ-ӨcosӨ) = -cotӨ[x-a(cosӨ+ӨsinӨ)],Is it correct?
3)If line ax+by+c=0 is normal to curve xy=1 then
b)a>0 , b<0 might be true
c)a<0,b>0, might be true
My approach: I first solved these eqn. to get the point of intersection of these curves.
x1 = [-c+V(c^2-4ab)]/2 , x2= [-c-V(c^2-4ab)]/2
then I differentiated xy=1, y'|(x1,y1)=-y/x= -y1/x1, then equating the slopes of normal form both the equation ..a/b =x1/y1..then how to intepret the signs of a and b..
4)Let Q(x) =f(x)+f(1-x) and f''(x)<0 , 0≤ x≤ 1 , then
a)Q increases in (1/2,1)
b)Q decreases in (1/2,1) true
c)Q decreases in (0,1/2)
d)Q increases in (0,1/2) true
f' is decreasing so Q'(x)=f'(x)-f'(1-x)>0 on (1/2,1)