Class 12

Math

Calculus

Application of Derivatives

Let AP and BQ be two vertical poles at points A and B, respectively. If $AP=16m,BQ=22mandAB=20m$, then find the distance of a point R on AB from the point A such that $RP_{2}+RQ_{2}$is minimum.

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If $f(x)andg(x)$ are continuous functions in $[a,b]$ and are differentiable in$(a,b)$ then prove that there exists at least one $c∈(a,b)$ for which. $∣f(a)f(b)g(a)g(b)∣=(b−a)∣∣ f(a)f_{prime}(c)g(a)g_{prime}(c)∣∣ ,wherea<c<b˙$

Prove that $f(x)=x−sinx$ is an increasing function.

Displacement $s$ of a particle at time $t$ is expressed as $s=21 t_{3}−6t˙$ Find the acceleration at the time when the velocity vanishes (i.e., velocity tends to zero).

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 $f(6)˙$

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 $f(x)andg(x)$ be differentiable function in $(a,b),$ continuous at $aandb,andg(x)=0$ in $[a,b]˙$ Then prove that $g(c)f_{prime}(c)−f(c)g_{prime}(c)g(a)f(b)−f(a)g(b) =(g(c))_{2}(b−a)g(a)g(b) $

Find $c$ of Lagranges mean value theorem for the function $f(x)=3x_{2}+5x+7$ in the interval $[1,3]˙$

Let $f(x)={x_{3}−x_{2}+10x−5,x≤1−2x+(g)_{2}(b_{2}−2),x>1$ Find the values of $b$ for which $f(x)$ has the greatest value at $x=1.$