Given A(0,0) and B(x,y) wih `x in (0,1) " and " y gt 0`. Let the slope of line AB be `m_(1), "where" 0 lt m_(2) lt m_(1)`. If the are of triangle ABC can be expresses as `(m_(1) - m_(2)) f (x).` then the largest possible value of `f(x)` is

Given A (0, 0) and B (x,y) with x epsilon (0,1) and y>0 . Let the slope of the line AB equals m_1 Point C lies on the line x= 1 such that the slope of BC equals m_2 where 0< m_2< m_1 If the area of the triangle ABC can expressed as (m_1- m_2)f(x) , then largest possible value of f(x) is:

A line passes through (x_(1), y_(1)) " and " (h, k) . If slope of the line is m, show that k - y_(1) = m (h - x_(1)) .

If f(x)=x/(x^2+1) and f(A)={y:-1/2 le y lt 0} , then set A is

If y=e^(m sin^(-1)x) , then the value of [(d^(2)y)/(dx^(2))]_(x=0) is -

Suppose that the foci of the ellipse (x^(2))/(9)+(y^(2))/(5)=-1 are (f_(1),0) and (f_(2),0) where f_(1)gt0 and f_(2)lt0 . Let P_(1) and P_(2) be two parabolas with a common vertex at (0, 0) and with foci at (f_(1),0) and (2f_(2),0) respectively. Let T_(1) be a tangent to P_(1) which passes through (2f_(2),0) and T_(2) be a tangent to P_(2) which passes Through (f_(1),0) . If m_(1) is the slope of T_(1) and m_(2) is the slope of T_(2) , then the value of ((1)/(m_(1)^(2))+m_(2)^(2)) is -

Let f'(x)=(192x^3)/(2+sin^4pix) for all x in R with f(1/2)=0.If mleint_(1//2)^1f(x)dxleM , then the possible values of m and M are

If m_(1),m_(2) be the roots of the equation x^(2)+(sqrt(3)+2)x+sqrt(3)-1 =0 , then the area of the triangle formed by the lines y = m_(1)x,y = m_(2)x and y = 2 is

Let RR be the set of all real numbers and A={x in RR : 0 lt x lt 1} . Is the mapping f : A to RR defined by f(x) =(2x-1)/(1-|2x-1|) bijective ?

Let f(x) =M a xi mu m {x^2,(1-x)^2,2x(1-x)}, where 0lt=xlt=1. Determine the area of the region bounded by the curves y=f(x) ,x-axis ,x=0, and x=1.

Show that the area of the triangle formed by the lines y = m_(1) x + c_(1) , y = m_(2) x + c_(2) " and " x = 0 " is " ((c_(1) - c_(2))^(2))/(2|m_(1) - m_(2)|)