Application of Derivatives
If f:RRis a differentiable function such that fprime(x)>2f(x)forallxRandf(0)=1,then:f(x)is decreasing in (0,∞) fprime(x)<e2x∈(0,∞) f(x)="" is="" increasing="" in="" (0,∞)="" f(x)=>e2x∈(0,∞)>
The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.
LL′ is the latus sectum of the parabola y2=4axandPP′ is a double ordinate drawn between the vertex and the latus rectum. Show that the area of the trapezium PPprimeLL′ is maximum when the distance PP′ from the vertex is a/9.
If t is a real number satisfying the equation 2t3−9t2+30−a=0, then find the values of the parameter a for which the equation x+x1=t gives six real and distinct values of x .