Applications of Derivatives Test

Result

Applications of Derivatives Test - 10

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Question 1
1 / -0.25

The interval in which the function x³ increases less rapidly than 6x² + 15x + 5 is

A
(– ∞,– 1)

B
(–5, 1)

C
(–1, 5)

D
(5, ∞)
Question 2
1 / -0.25

The function is monotonically decreasing in

A
(2, ∞)

B
(0, 1)

C
(0, 1) and (2, ∞)

D
(–∞, ∞)

Solution

f(x)= {−(x −1)/x² x <1, x(x −1)/x² x ≥1}
f '(x)= {(x −2)/x³ x <1, −(x −2)/x³ x ≥1}
x <1, if f ′(x)<0 (for f(x) to be monotonically decreasing
⇒(x −2)/x³ <0
⇒x ∈(0,2)
But x <1 ⇒x ∈(0,1)
For x ≥1, if f ′(x)<0
⇒−(x −2)]/x³ <0
⇒(x −2)/x³ >0
⇒x ∈(−∞,0)∪(2,∞)
But, x ≥1 ⇒x ∈(2,∞)
Hence, x ∈(0,1)∪(2,∞)
Question 3
1 / -0.25

If y = (a + 2) x³ – 3ax² + 9ax – 1 decreases monotonically x ∈R then `a 'lies in the interval

A
(– ∞, – 3]

B
(– ∞, _ 2) ∪(–2, 3)

C
(–3, ∞)

D
None of these
Question 4
1 / -0.25

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

A
(1, ∞)

B
(1/e, ∞)

C
[e, ∞)

D
(0, 1) and (1, ∞)

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
1 / -0.25

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

A
x >0

B
x >–1

C
–1

D
Null set

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
1 / -0.25

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

A

B

C

D

Solution

f ’(x) >0
=>f(x) is increasing
f ’’(x) <0 =>f(x) is convex
Question 7
1 / -0.25

If the point (1, 3) serves as the point of inflection of the curve y = ax³ + bx² then the value of `a 'and `b 'are

A
a = 3/2 &b = –9/2

B
a = 3/2 &b = 9/2

C
a = –3/2 &b = –9/2

D
a = –3/2 &b = 9/2
Question 8
1 / -0.25

The function f(x) = x³ – 6x² + ax + b satisfy the conditions of Rolle 's theorem in [1, 3]. The value of a and b are

A
11, –6

B
–6, 11

C
–11, 6

D
6, –11
Question 9
1 / -0.25

If f(x) = ; g(x) = for a >1, a ¹1 and x ÎR, 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)).

A
`h 'is even and increasing

B
`h 'is odd and decreasing

C
`h 'is even and decreasing

D
`h 'is odd and increasing
Question 10
1 / -0.25

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

A
f '(x) = 0 has four roots

B
Three roots of f '(x) = 0 lie in (4, 5) ∪(5, 6) ∪(6, 7)

C
The equation f '(x) = 0 has only one real root

D
Three roots of f '(x) = 0 lie in (3, 4) ∪(4, 5) ∪(5, 6)
Question 11
1 / -0.25

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

A
f(x) is an increasing function

B
f(x) is a decreasing function

C
|f(x)| is an increasing function

D
|f(x)| is a decreasing function