Class 12

Math

Calculus

Application of Derivatives

Let f be a function defined on [a, b] such that $f_{prime}(x)>0$, for all $x∈(a,b)$. Then prove that f is an increasing function on (a, b).

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The side of an equilateral triangle are increasing at the rate of $2cm/s$. Find the rate at which the area is increasing when the side is $10cm$.

Show that the maximum value of $x_{1/x}$ is $e_{1/e}$

Show that the function given by $f(x)=3x+17$ is strictly increasing on $R$.

Show that $(x+x1 )$ has a maximum and minimum, but the maximum value is less than the minimum value.

Find the intervals in which the function $f$ given by $f(x)=2x_{2}−3x$ is (a) strictly increasing (b) strictly decreasing.

The side of a square is increasing at the rate of $0.2cm/s$. Find the rate of increase of the perimeter of the square.

Discuss the applicability of Rolle's theorem when:$f(x)=cosx1 $ on $[−1,1]$

A man is moving away from a $40m$ high tower at a speed of $2m/s$. Find the rate at which the angle of elevation of the top of the tower is changing when he is at a distance of $30$ metres from the foot of the tower. Assume that the eye of the man is $1.6m$ from the ground.