Application of Derivatives
A balloon, which always remains spherical, has a variable diameter 23(2x+1).Find the rate of change of its volume with respect to x.
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Find the points of local maxima or local minima and the corresponding local maximum and minimum values of the following function:f(x)=x4−62x2+120x+9
An open tank with a square base and vertical sides is to be constructed from a metal sheet so as to hold a given quantity of water. Show that the cost of the material will be least when depth of the tank is half of its width.
Let I be any interval disjoint from [−1,1]. Prove that the function f given byf(x)=x+x1 is strictly increasing in interval disjoint from I.
Verify Lagranges's mean-value theorem for the following function:f(x)=sinx on [2π,25π]
Discuss the applicability of Rolle's theorem when:f(x)=(x−1)(2x−3), where 1≤x≤3
Show that the tangents to the curve y=7x3+11 at the points where x=2 and x=−2 are parallel.
Find the point on the curve y=x3−11x+5 at which the tangent is y=x−11.
The radius of a circle is increasing uniformly at the rate of 0.3 centimetre per second. At what rate is the area increasing when the radius is 10 cm? (Taken π=3.14.)