Consider the functionf(x) = 3x4 – 20x3 &ndas

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 Multiple Choice QuestionsMultiple Choice Questions

21.

What is the minimum value of a2cos2x + b2sin2x, where a > 0 and b > 0 ?

  • (a + b)2

  • (a - b)2

  • a2  + b2

  • a2 + b2


22.

If fx = x23 - 5x22 + 6x + 7 increases in the interval T and decreases in the interval S, then which one of the following is correct ?

  • T = - , 2  3,  and S = 2, 3

  • T = φ and S =  - , 

  • T =  - ,  and S = φ

  • T = 2, 3 and S = - , 2  3, 


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23.

Consider the function

f(x) = 3x4 – 20x3 – 12x2 + 288x + 1

In which one of the following intervals is the function decreasing ?

  • ( - 2, 3)

  • (3, 4)

  • (4, 6)

  • No


B.

(3, 4)

Given that f(x) = 3x– 20x3 – 12x2 + 288x + 1

Now, we have to differentiate f(x) w.r.t x

If f’(x) > 0 for particular value of x then f(x) is increasing

If f’(x) < 0 for particular value of x then f(x) is decreasing.

Here, we have to check the range of x for an increasing f(x)

Then f’(x) > 0

f’(x) = 12x3 60x2 24x +288

now, we will have to find the roots of the equation, but we know that it is a cubic polynomial hence, it is very difficult to find out the roots of a cubic polynomial

hence for paper point of view.

So, we will go through the option and we can clearly see that the options are allhave different range,

Not common to anyone, so we can assume a value from the given ranges in the option and check whether the f’(x) is greater or less than zero.Here we know that the function f(x) is increasing in (- 2, 3) hence, we have to check option B,

And we also knw that the f’(x) is having a curve with alternate positive and negative region,

Hence we can clearly say that the f’(x) < 0 in (3, 4).

So, option B is correct


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24.

Consider the function

f(x) = 3x4 – 20x3 – 12x2 + 288x + 1

In which one of the following intervals is the function increasing ?

  • ( - 2, 3)

  • (3, 4)

  • (4, 6)

  • (6, 9)


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25.

Let l be the length and b be the breadth of a rectangle such that l x b = k. What is the maximum area of the rectangle ?

  • 2k2

  • k2

  • k22

  • k24


26.

What is the minimum value of 3cosA + π3 where A ∈ R ?

  • - 3

  • - 1

  • 0

  • 3


27.

The radius of a circle is increasing at the rate of 0.7 cm/sec. What is the rate of increase of its circumference ?

  • 4.4 cm/sec

     

  • 8.4 cm/sec

  • 8.8 cm/sec

  • 15.4 cm/sec


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