Class 12

Math

Calculus

Application of Derivatives

Find the maximum and the minimum values, if any, of the function f given by$f(x)=x_{2},x∈R$.

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Assuming the petrol burnt (per hour) in driving a motor boat veries as the cube of its velocity, show that the most economical speed when going against the current of $c$ miles per hour is $(23c )$ miles per hour.

Water is dropped at the rate of 2 $m_{3}$/s into a cone of semi-vertical angle is $45_{∘}$ . If the rate at which periphery of water surface changes when the height of the water in the cone is 2m is d. Then the value of 5d is _____ m/sec

Let $f:[2,7]0,∞ $ be a continuous and differentiable function. Then show that $(f(7)−f(2))3(f(7))_{2}+(f(2))_{2}+f(2)f(7) =5f_{2}(c)f_{prime}(c),$ where $c∈[2,7]˙$

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. $∣f(a)f(b)g(a)g(b)∣=(b−a)∣∣ f(a)f_{prime}(c)g(a)g_{prime}(c)∣∣ ,wherea<c<b˙$

For the curve $y=f(x)$ prove that (lenght n or mal)^2/(lenght or tanght)^2

Find the cosine of the angle of intersection of curves $f(x)=2_{x}(g)_{e}xandg(x)=x_{2x}−1.$

If the curves $ay+x_{2}=7andx_{3}=y$ cut orthogonally at $(1,1)$ , then find the value $a˙$

A private telephone company serving a small community makes a profit of Rs. 12.00 per subscriber, if it has 725 subscribers. It decides to reduce the rate by a fixed sum for each subscriber over 725, thereby reducing the profit by 1 paise per subscriber. Thus, there will be profit of Rs. 11.99 on each of the 726 subscribers, Rs. 11.98 on each of the 727 subscribers, etc. What is the number of subscribers which will give the company the maximum profit?