Class 12

Math

Calculus

Application of Derivatives

Find local maximum and local minimum values of the function f given by$f(x)=3x_{4}+4x_{3}−12x_{2}+12$.

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Find the locus of point on the curve $y_{2}=4a(x+as∈ax )$ where tangents are parallel to the axis of $x˙$

Find the length of the tangent for the curve $y=x_{3}+3x_{2}+4x−1$ at point $x=0.$

Two towns $AandB$ are 60 km apart. A school is to be built to serve 150 students in town $Aand50$ students in town $B˙$ If the total distance to be travelled by 200 students is to be as small as possible, then the school should be built at (a)town B (b)town A (c)45 km from town A (d)45 km from town B

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 $f(x)$ be defined as $f(x)={tan_{−1}α−5x_{2},0<x<1and−6x,x≥1$ if $f(x)$ has a maximum at $x=1,$ then find the values of $α$ .

The lateral edge of a regular rectangular pyramid is $acmlong˙$ The lateral edge makes an angle $α$ with the plane of the base. Find the value of $α$ for which the volume of the pyramid is greatest.

The sum of the perimeter of a circle and square is k, where k is some constant. Prove that the sum of their areas is least when the side of square is double the radius of the circle.

Find the maximum and minimum values of the function $y=(g)_{e}(3x_{4}−2x_{3}−6x_{2}+6x+1)∀x∈(0,2)$ Given that$(3x_{4}−2x_{3}−6x_{2}+6x_{2}+6x+1)>0Ax∈(0,2)$