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QUESTION: 1

The interval in which the function x^{3} increases less rapidly than 6x^{2} + 15x + 5 is

Solution:

QUESTION: 2

The function is monotonically decreasing in

Solution:

f(x)= {−(x−1)/x^{2} x<1, x(x−1)/x^{2} x≥1}

f'(x)= {(x−2)/x^{3} x<1, −(x−2)/x^{3} x≥1}

x<1, if f′(x)<0 (for f(x) to be monotonically decreasing

⇒ (x−2)/x^{3}<0

⇒x∈(0,2)

But x<1 ⇒ x∈(0,1)

For x≥1, if f′(x)<0

⇒ −(x−2)]/x^{3}<0

⇒(x−2)/x^{3} > 0

⇒x∈(−∞,0)∪(2,∞)

But, x≥1 ⇒x∈(2,∞)

Hence, x∈(0,1)∪(2,∞)

QUESTION: 3

If y = (a + 2) x^{3} – 3ax^{2} + 9ax – 1 decreases monotonically x ∈ R then `a' lies in the interval

Solution:

QUESTION: 4

The true set of real values of x for which the function, f(x) = x *l*n x – x + 1 is positive is

Solution:

f(x)=xlnx−x+1

f(1)=0

On differentiating w.r.t x, we get

f′(x)=x*1/x+lnx−1

=lnx

Therefore,f′(x)>0 for allx∈(1,∞)

f′(x)<0 for allx∈(0,1)

lim x→0 f(x)=1

f(x)>0

for allx∈(0,1)∪(1,∞)

QUESTION: 5

The set of all x for which *l*n (1 + x) ≤ x is equal to

Solution:

f(x) = ln(1+x) - x ≤ 0

f(x) = ln(1+x) - x

f’(x) = 1/(1+x) - 1

= (1 - 1 - x)/(1+x)

= -x/(1+x)

f’(x) ≤ 0

-x/(1+x) ≤ 0

0 ≤ x /(1+x)

for(x = 2) (-2)/(1-2)

= 2 > 0, therefore x > - 1

QUESTION: 6

The curve y = f(x) which satisfies the condition f'(x) > 0 and f"(x) < 0 for all real x, is

Solution:

f’(x) > 0

=> f(x) is increasing

f’’(x) < 0 => f(x) is convex

QUESTION: 7

If the point (1, 3) serves as the point of inflection of the curve y = ax^{3} + bx^{2} then the value of `a' and `b' are

Solution:

QUESTION: 8

The function f(x) = x^{3} – 6x^{2} + ax + b satisfy the conditions of Rolle's theorem in [1, 3]. The value of a and b are

Solution:

QUESTION: 9

If f(x) = ; g(x) = for a > 1, a ¹ 1 and x Î R, where {*} & [*] denote the fractional part and integral part functions respectively, then which of the following statements holds good for the function h(x), where (*l*n a) h(x) = (*l*n f(x) +*l*n g(x)).

Solution:

QUESTION: 10

Let f(x) = (x – 4) (x – 5) (x – 6) (x – 7) then,

Solution:

QUESTION: 11

Given that f is a real valued differentiable function such that f(x) f'(x) < 0 for all real x, it follows that

Solution:

QUESTION: 12

If f(x) = ; g(x) = where 0 < x < 1, then

Solution:

QUESTION: 13

f : R → R be a differentiable function x ∈ R. If tangent drawn to the curve at any point x ∈ (a, b) always lie below the curve, then

Solution:

QUESTION: 14

A value of C for which the conclusion of Mean Value Theorem holds for the function f(x) = log_{e} x on the interval [1, 3] is

Solution:

QUESTION: 15

The function f(x) = x(x + 3) e^{–x/2} satisfies all the conditions of Rolle’s theorem in [–3, 0]. The value of c which verifies Rolle’s theorem, is

Solution:

f(x) = x (x + 3)e–x/2 is continuous in [–3, 0]

and f′(x)=(x2 + 3x) (– 1/2)e-x/2 + (2x + 3)e–x/2

= – 1/2(x2 – x – 6)e–x/2

Therefore f′(x) exists (i.e., finite) for all x

Also f(–3) = 0, f (0) = 0

So that f(–3) = f(0)

Hence all the three conditions of the theorem are satisfied.

Now consider f′(c)=0

i.e., –1/2(c2 – c – 6)e–c/2 = 0

c2 – c – 6 = 0

(c + 2) (c – 3) = 0 c = 3 or –2

Hence there exists – 2 ∈ (–3, 0) such that

f′(–2) = 0

QUESTION: 16

x^{3} – 3x^{2} –_{ }9x + 20 is

Solution:

QUESTION: 17

f(x) = x^{2} - x sin x is

Solution:

QUESTION: 18

The number of values of `c' of Lagrange's mean value theorem for the function,

f(x) = (x – 1) (x – 2) (x – 3), x ∈ (0, 4) is

Solution:

QUESTION: 19

If f(x) and g(x) are differentiable in [0, 1] such that f(0) = 2, g(0) = 0, f(1) = 6, g(1) = 2, then Rolle's theorem is applicable for which of the following

Solution:

QUESTION: 20

Equation 3x^{2} + 4ax + b = 0 has at least one root in (0, 1) if

Solution:

QUESTION: 21

Function for which LMVT is applicable but Rolle's theorem is not

Solution:

QUESTION: 22

LMVT is not applicable for which of the following ?

Solution:

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