Class 12

Math

Calculus

Application of Derivatives

Find the maximum and minimum values, if any, of the following functions given by (i) $f(x)=(2x−1)_{2}+3$ (ii) $f(x)=9x_{2}+12x+2$(iii) $f(x)=−(x−1)_{2}+10$ (iv) $g(x)=x_{3}+1$

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Using Lagranges mean value theorem, prove that $bb−a <g(ab )<ab−a =a,$where $0<a<b˙$

Discuss the extremum of $f(x)=2x_{3}−3x_{2}−12x+5$ for $x∈[−2,4]$ and the find the range of $f(x)$ for the given interval.

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

For the curve $y=4x_{3}−2x_{5},$find all the points at which the tangent passes through the origin.

Find the equation of the tangent to the curve $(1+x_{2})y=2−x,$ where it crosses the x-axis.

For the curve $y=a1n(x_{2}−a_{2})$ , show that the sum of length of tangent and sub-tangent at any point is proportional to product of coordinates of point of tangency.

Let $f$ defined on $[0,1]$ be twice differentiable such that $∣f(x)∣≤1$ for $x∈[0,1]$. if $f(0)=f(1)$ then show that $∣f_{′}(x)<1$ for all $x∈[0,1]$.

The acute angle between the curves $y=∣∣ x_{2}−1∣∣ $and $y=∣∣ x_{2}−3∣∣ $ at their points of intersection when when x> 0, is