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NCERT Class 12 Maths Chapter 6: Applications Of Derivatives 29 Questions Solved

NCERT Solutions for Class 12 Maths Chapter 6 Applications Of Derivatives — Exercise 6.3

Click any question below for its step-by-step NCERT textbook solution, concept hints, and worked explanation.

Find maximum/minimum values, if any: (i) (2x−1)²+3; (ii) 9x²+12x+2; (iii) −(x−1)²+10; (iv) x³+1.

Find maximum/minimum values: (i) |x+2|−1; (ii) −|x+1|+3; (iii) sin 2x+5; (iv) |sin 4x+3|; (v) x+1, −1<x<1.

Find all local extrema and their values: (1) \(x^{2}\); (2) \(x^{3} - 3 x\); (3) \(\sin{\left(x \right)} + \cos{\left(x \right)}\), 0…

Prove eˣ, log x and x³+x²+x+1 have no maxima or minima on their natural domains.

Find absolute extrema: (i) x³ on [−2,2]; (ii) sin x+cos x on [0,π]; (iii) 4x−x²/2 on [−2,9/2]; (iv) (x−1)²+3 on [−3,1].

Find the maximum profit for p(x)=41−72x−18x².

Find absolute extrema of 3x⁴−8x³+12x²−48x+25 on [0,3].

Where on [0,2π] does sin 2x attain its maximum?

Find the maximum of sin x+cos x.

Find the maximum of 2x³−24x+107 on (i) [1,3], (ii) [−3,−1].

x⁴−62x²+ax+9 attains its maximum on [0,2] at x=1. Find a.

Find maximum and minimum of x+sin 2x on [0,2π].

Find two numbers with sum 24 and maximum product.

Find positive x,y with x+y=60 maximizing xy³.

Find positive x,y with x+y=35 maximizing x²y⁵.

Find positive numbers with sum 16 and minimum sum of cubes.

Cut equal corner squares from a 18 cm × 18 cm tin sheet and fold an open box. Find the cut size maximizing volume.

Cut equal corner squares from a 45 cm × 24 cm tin sheet and fold an open box. Find the cut size maximizing volume.

Show that a square has maximum area among rectangles inscribed in a fixed circle.

Show that a right circular cylinder with fixed surface area and maximum volume has height equal to its base diameter.

For a closed cylindrical can of volume 100 cm³, find dimensions minimizing surface area.

Cut 28 m wire into a square perimeter and circle circumference to minimize their combined area.

Prove the largest cone inscribed in a sphere of radius R has volume 8/27 of the sphere volume.

Show that the cone of given volume and least curved surface has height √2 times base radius.

Show that a cone with fixed slant height and maximum volume has semi-vertical angle tan⁻¹√2.

Show that a cone of fixed total surface area and maximum volume has semi-vertical angle sin⁻¹(1/3).

Which listed point on x²=2y is nearest (0,5)?

Find the minimum over real x of (1−x+x²)/(1+x+x²).

Find the maximum of [x(x−1)+1]^(1/3) on 0≤x≤1.