Class 12

Math

Calculus

Application of Derivatives

Points on the curve $f(x)=1−x_{2}x $ where the tangent is inclined at an angle of $4π $ to the x-axis are (0,0) (b) $(3 ,−23 )$ $(−2,32 )$ (d) $(−3 ,23 )$

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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.

Find the value of $a$ if $x_{3}−3x+a=0$ has three distinct real roots.

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 $cf_{prime}(c)−f(c)=0$ $cf_{prime}(c)+cf(c)=0$ $f_{prime}(c)−cf(c)=0$ $cf_{prime}(c)+f(c)=0$

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 $274 πh_{3}tan_{2}α˙$

A spherical iron ball 10cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $50cm_{3}/m∈$ . 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.

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?