Show that the semi-vertical angle of the cone of the maximum volume and of given slant height is \(tan^{-1}\sqrt{2}.\)
Show that the right circular cone of least curved surface and given volume has an altitude equal to\(\sqrt{2}\) time the radius of the base.
Find the absolute maximum value and the absolute minimum value of the following functions in the given intervals: (i)f(x)=x3,x∈[-2,2] (ii) f(x)=sin x+cos x,x∈[0,π] (iii) f(x)=4x-1/2x2,x∈[-2,\(\frac{9}{2}\)] (iv) f(x)=(x-1)2+3,x∈[-3,1]
Prove that the following functions do not have maxima or minima: (i) f(x) = ex (ii) g(x) = logx (iii) h(x) = x3 + x2+x+1
Find the maximum and minimum values, if any, of the following functions given by (i) f(x) = (2x − 1)2 + 3 (ii) f(x) = 9x2+12x+2 (iii) f(x) = −(x − 1)2+ 10 (iv) g(x) = x3 +1
Find the approximate change in volume V of a cube of side x meters caused by increasing side by 1%.
Find the approximate value of f (5.001), where f (x) = x3 − 7x2 + 15
Find the approximate value of f (2.01), where f (x) = 4x2+5x + 2
The anti derivative of \(\bigg(\sqrt x+\frac{1}{\sqrt x}\bigg)\) equals
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
Find the equation of the normals to the curve y = x3 + 2x + 6 which are parallel to the line x + 14y + 4 = 0.
The slope of the normal to the curve y = 2x2 + 3 sin x at x = 0 is
Find the equations of the tangent and normal to the hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}\)=1at the point (x0,y0).
Prove that the curves x = y2 and xy = k cut at right angles if 8k2 = 1. [Hint: Two curves intersect at right angle if the tangents to the curves at the point of intersection are perpendicular to each other.]