Application of Derivatives
Let f,g and h be real-valued functions defined on the interval [0,1] by f(x)=ex2+e−x2 , g(x)=xex2+e−x2 and h(x)=x2ex2+e−x2. if a,b and c denote respectively, the absolute maximum of f,g and h on [0,1] then
Show that the normal at any point θto the curvex=acosθ+aθsinθ,y=asinθ−aθcosθis at a constant distance from the origin.
Find intervals in which the function given by f(x)=103x4−54x3−3x2+536+11is (a) strictly increasing (b) strictly decreasing.
The radius of a circle is increasing uniformly at the rate of 3 cm/s. Find the rate at which the area of the circle is increasing when the radius is 10 cm.
Let AP and BQ be two vertical poles at points A and B, respectively. If AP=16m,BQ=22mandAB=20m, then find the distance of a point R on AB from the point A such that RP2+RQ2is minimum.
Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is 34r.
Find the absolute maximum and minimum values of a function f given by f(x)=2x3−15x2+36x+1on the interval (3.02),[1,5].
A water tank has the shape of an inverted right circular cone with its axis vertical and vertex lowermost. Its semi-vertical angle is tan−1(0.5). Water is poured into it at a constant rate of 5 cubic metre per hour. Find the rate at which the level of the water is rising at the instant when the depth of water in the tank is 4m.