Class 12

Math

Calculus

Application of Derivatives

If the slope of line through the origin which is tangent to the curve $y=x_{3}+x+16$ is $m,$ then the value of $m−4$ is____.

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Find the maximum value and the minimum value and the minimum value of $3x_{4}−8x_{3}+12x_{2}−48x+25$ on the interval $[0,3]˙$

If the tangent at any point $(4m_{2},8m_{2})$ of $x_{3}−y_{2}=0$ is a normal to the curve $x_{3}−y_{2}=0$ , then find the value of $m˙$

Tangent of an angle increases four times as the angle itself. At what rate the sine of the angle increases w.r.t. the angle?

In an acute triangle $ABC$ if sides $a,b$ are constants and the base angles $AandB$ vary, then show that $a_{2}−b_{2}sin_{2}A dA =b_{2}−a_{2}sin_{2}B dB $

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $R$ is $3 2R $

Show that the straight line $xcosα+ysinα=p$ touches the curve $xy=a_{2},$ if $p_{2}=4a_{2}cosαsinα˙$

Find the cosine of the angle of intersection of curves $f(x)=2_{x}(g)_{e}xandg(x)=x_{2x}−1.$

Let $f$ be differentiable for all $x,$ If $f(1)=−2andf_{prime}(x)≥2$ for all $x∈[1,6],$ then find the range of values of $f(6)˙$