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Applications of Derivatives

Applications of Derivatives

by Vibhavari Dasagi -
Number of replies: 4

Can someone please clear these doubts?

1. P(x,y) is a point on the curve y2 = 2x3 , such that the tangent is perpendicular to the line 4x-3y+2=0. Then, 1/x - 1/y =__________

a) 24

b) 50

c) 8

d) 4

Ans. : a)

2. Let f be the function defined by f(x) = cos x - 1 + x2/2. Then

a) f(x) is increasing on the real line

b) f(x) is decreasing on the real line

c) f(x) increases in (-infinity,0] and decreases in [0, infinity)

d) f(x) decreases in (-infinity,0] and increases in [0, infinity)

Ans. : d)

Please give the explanation as well. Thnx!

In reply to Vibhavari Dasagi

Re: Applications of Derivatives

by Jeffrey John -
(a) the first one is quite simple.
y 2 =2x3.............................(1)
differntiate w.r.t.x,
2y.(dy/dx)=6x2.
so (dy/dx)=(3x2)/y.
since the tangent at P is perpendicular to 4x-3y+2=0, we get,
((3x2)/y)*(4/3) = -1. (since m1*m2= -1 for perpendicular lines)
which gives 4x2=-y.................(2)
Now consider equation (1).
y 2 =2x3 this can be written as,
y2 =4x2 * (x/2).
or y2 =-y * (x/2) ( from (2) )
or x = -2y.
substituting this value of x in (1), we get y=0 or y=(-1/16).
so x=0 or x= (1/8).
for x=0 and y=0, (1/x)-(1/y) is not defined.
for x=(1/8) and y=(-1/16), we get (1/x)-(1/y) = 24.
hence the correct answer is (a) 24.
In reply to Vibhavari Dasagi

Re: Applications of Derivatives

by Rakesh Arigala -

2 question ans should be d.

f(x)=cosx-1+x2/2

differentiate w.r.t.x

f'x= -sinx+x

increasing if f'(x)>0and decreasing if f'(x)<0,in both cases option d satisfies.

if increasing replace x by pi and if decreasing replace x by -pi.