Application of Derivatives
Points on the curve f(x)=1−x2x where the tangent is inclined at an angle of 4π to the x-axis are (0,0) (b) (3,−23) (−2,32) (d) (−3,23)
Let y=f(x) be a polynomial of odd degree (≥3) with real coefficients and (a, b) be any point. Statement 1: There always exists a line passing through (a,b) and touching the curve y=f(x) at some point. Statement 2: A polynomial of odd degree with real coefficients has at least one real root.
If f is a continuous function on [0,1], differentiable in (0, 1) such that f(1)=0, then there exists some c∈(0,1) such that cfprime(c)−f(c)=0 cfprime(c)+cf(c)=0 fprime(c)−cf(c)=0 cfprime(c)+f(c)=0
Let g(x)=(f(x))3−3(f(x))2+4f(x)+5x+3sinx+4cosx∀x∈R˙ Then prove that g is increasing whenever is increasing.
Show that height of the cylinder of greatest volume which can be inscribed in a right circular cone of height h and semi vertical angle is one-third that of the cone and the greatest volume of cylinder is 274πh3tan2α˙
A spherical iron ball 10cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of 50cm3/m∈ . When the thickness of ice is 5cm, then find the rate at which the thickness of ice decreases.
Let f:[0,∞)0,∞andg:[0,∞)0,∞ be non-increasing and non-decreasing functions, respectively, and h(x)=g(f(x))˙ If fandg are differentiable functions, h(x)=g(f(x))˙ If fandg are differentiable for all points in their respective domains and h(0)=0, then show h(x) is always, identically zero.