Ch06. Applications of Derivatives

Assignment – 1

• Question 1. An open box with square base is to be made out of a given iron sheet of area 27 sq. meter; show that the maximum volume of the box is 13.5 cubic meters.
• Question 2. Prove that the surface area of solid cuboid of a square base and given volume is minimum, when it is a cube.
• Question 3. A given quantity of metal is to be cast half cylindere., with a rectangular base and semicircular ends. Show that the total surface area is minimum when the ratio of the length of cylinder to the diameter of its semicircular ends is {\displaystyle \pi :(\pi +2)} .
• Question 4. What is the largest possible area for a right triangle whose hypotenuse is 5 cm long?
• Question 5. Show that the line {\displaystyle \frac{x}{a}+\frac{y}{b}=1} touches the curve {\displaystyle y=b{{e}^{\frac{-x}{a}}}} at the point, where the curve intersects the axis of y.
• Question 6. Show that the curves {\displaystyle y={{a}^{x}}} and {\displaystyle y={{b}^{x}},\,\,\,\,\,\,a>b>0} intersect at an angle of {\displaystyle {{\tan }^{-1}}\left( \left| \frac{\log \left| \frac{a}{b} \right|}{1+\log a\log b} \right| \right)} .
• Question 7. Prove that the least perimeter of an isosceles triangle in which a circle of radius r can be inscribed is {\displaystyle 6\sqrt{3}r} .
• Question 8. An isosceles triangle of vertical angle 2θ is inscribed in a circle of radius a. Show that the area of triangle is maximum when {\displaystyle \theta =\frac{\pi }{6}} .

Assignment – 2

Find the intervals in which the following functions increasing or decreasing:

• Question 1. {\displaystyle f(x)={{x}^{4}}-8{{x}^{3}}+22{{x}^{2}}-24x+5}
• Question 2. {\displaystyle f(x)=2{{x}^{2}}-lnx}
• Question 3. {\displaystyle f(x)=\frac{1-x+{{x}^{2}}}{1+x+{{x}^{2}}}}
• Question 4. {\displaystyle f(x)=\frac{x}{{{x}^{2}}+1}}
• Question 5. {\displaystyle f(x)=2log(x-2)-{{x}^{2}}+4x+1}
• Question 6. {\displaystyle f(x)=si{{n}^{4}}x+co{{s}^{4}}x,\,x\in \left[ 0,\,\,\frac{\pi }{2} \right]}
• Question 7. Prove that the function f given by {\displaystyle f(x)=\log |\cos x|} is decreasing on {\displaystyle \left( 0,\,\,\frac{\pi }{2} \right)} and increasing on {\displaystyle \left( \frac{3\pi }{2},\,\,2\pi \right)} .   (NCERT Exercise 6.2 Q17)

Assignment – 3

• Show that the local maximum value of {\displaystyle x+\frac{1}{x}} is less than local minimum value.
• Find the difference between the greatest and the least values of the function {\displaystyle f(x)=sin2x-x,} on {\displaystyle \left[ \frac{-\pi }{2},\,\,\frac{\pi }{2} \right]} .
• Find the maximum and minimum values of             {\displaystyle f(x)=secx+logco{{s}^{2}}x,\,\,\,0<x<2\pi } .
• Find the area of greatest rectangle that can be inscribed in an ellipse {\displaystyle \frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1} .
• An open box with the square base is to be made of a given quantity of cardboard of area {\displaystyle {{c}^{2}}} . Show that the maximum volume of the box is {\displaystyle \frac{{{c}^{3}}}{6\sqrt{3}}} cubic units.
• The sum of the surface areas of a rectangular parallelepiped with sides {\displaystyle x,\,\,2x} & {\displaystyle \frac{x}{3}} and a sphere is given to be constant. Prove that the sum of their volumes is minimum if x is equal to three times the radius of the sphere. Also, find the minimum value of the sum of their volumes.