CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. M.P.Keshari

Application of derivatives

Q.3. The volume of a cube is increasing at the rate of 7 cubic centimeters per second. How fast is the surface area of the cube increasing when the length of an edge is 12 centimeters?

Solution :

Let the edge of the cube be a, the volume be V and surface area be S.
Now, dV/dt = 7 cm/sec
=> d/dt(a³) = 7
=> 3a² da/dt = 7
=> da/dt = 7/(3a²).
Now, S = 6 a² => dS/dt = 12 a da/dt
= 12 a.(7/3a²) = 28/a
=> dS/dt = 28/12 [where a = 12 cm]
= 7/3 sq. cm/sec. [Ans.]

6.2. Increasing and Decreasing Function.

Q.1. Find the intervals in which the function f(x) = x³ – 12 x² + 36 x + 17 is

1. increasing,
2. decreasing.

Solution :

We have, f(x) = x³ – 12 x² + 36 x + 17
Therefore, f’(x) = 3x² – 24 x + 36
= 3(x – 2)(x – 6).

1. For f(x) to be increasing, f’(x) > 0
   Or, 3(x – 2)(x – 6) > 0
   Either x – 2 > 0 and x – 6 > 0 or x – 2 < 0 and x – 6 < 0
   Either x > 2 and x > 6 or x < 2 and x < 6
   Or, x ε ] – ∞, 2[ U ] 6, ∞ [ . [Ans.]
   
   
2. For x to be decreasing, f’(x) < 0
   Or, 3(x – 2)(x – 6) < 0
   Either x – 2 < 0 and x – 6 > 0 or x – 2 > 0 and x – 6 < 0
   Either x < 2 and x > 6 or x > 2 and x < 6
   Or, x > 2 and x < 6
   Or, x ε ]2, 6[ [Ans.]

Q.2. Find the intervals in which the function f(x) = 2x3 – 9x2 + 12x + 15 is (i) increasing and (ii) decreasing.

Solution :

We have, f(x) = 2x³ – 9x² + 12x + 15
f’(x) = 6x² – 18x + 12
= 6(x² – 3x + 4)
= 6(x – 1)(x – 2).
For increasing and decreasing function f’(x) = 0
=> 6(x – 1)(x – 2) = 0
=> x = 1, 2.
Therefore, disjoint intervals on real number line are (– ∞, 1), (1, 2), (2, ∞)

Therefore, f(x) is increasing in ( – ∞, 1), (2, ∞) and decreasing in (1, 2). [Ans.]

Paper By Mr. M.P.Keshari
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Ph No. : 09434150289