Class 12

Math

Calculus

Application of Derivatives

Find the points on the curve $y=x_{3}$at which the slope of the tangent is equal to the y-coordinate of the point.

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Discuss the extremum of $f(x)=sinx+21 sin2x+31 sin3x,0≤x≤π˙$

If the curve $y=ax_{2}−6x+b$ pass through $(0,2)$ and has its tangent parallel to the x-axis at $x=23 ,$ then find the values of $aandb˙$

Let $x$ be the length of one of the equal sides of an isosceles triangle, and let $θ$ be the angle between them. If $x$ is increasing at the rate (1/12) m/h, and $θ$ is increasing at the rate of $180π $ radius/h, then find the rate in $m_{3}$ / $h$ at which the area of the triangle is increasing when $x=12mandthη=π/4.$

Let $P(x)$ be a polynomial with real coefficients, Let $a,b∈R,a<b,$ be two consecutive roots of $P(x)$ . Show that there exists $c$ such that $a≤c≤bandP_{prime}(c)+100P(c)=0.$

If the line $xcosθ+ysinθ=P$ is the normal to the curve $(x+a)y=1,$ then show $θ∈(2nπ+2π ,(2n+1)π)∪(2nπ+23π ,(2n+2)π),n∈Z$

At the point $P(a,a_{n})$ on the graph of $y=x_{n},(n∈N)$ , in the first quadrant, a normal is drawn. The normal intersects the $y−aξs$ at the point $(0,b)$ . If $(lim)_{a0}=21 ,$ then $n$ equals _____.

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 $274 πh_{3}tan_{2}α˙$

Find the minimum value of $∣x∣+∣∣ x+21 ∣∣ +∣x−3∣+∣∣ x−25 ∣∣ ˙$