Applications of Derivatives Test

Result

Applications of Derivatives Test - 11

Score
-
out of -
Rank
-
out of -

TIME Taken - -

SHARING IS CARING

If our Website helped you a little, then kindly spread our voice using Social Networks. Spread our word to your readers, friends, teachers, students & all those close ones who deserve to know what you know now.

Weekly Quiz Competition

Question 1
1 / -0.25

If f(x) = ; g(x) = where 0

A
both `f 'and `g 'are increasing functions

B
f 'is decreasing &`g 'is increasing function

C
'f 'is increasing &`g 'is decreasing function

D
both `f '&`g 'are decreasing function
Question 2
1 / -0.25

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

A
f '(x) >0 f "(x) <0   x ∈ (a, b)

B
f '(x) <0 f "(x) <0   x ∈(a, b)

C
f '(x) >0 f "(x) >0   x ∈(a, b)

D
None of these
Question 3
1 / -0.25

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

A
2 log₃ e

B
log_e 3

C
log₃ e

D
log_e 3
Question 4
1 / -0.25

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

A
0

B
-2

C
-1

D
3

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 5
1 / -0.25

x³ – 3x² – 9x + 20 is

A
–ve for x <4

B
+ve for x >4

C
–ve for x ∈(0, 1)

D
–ve for x ∈(–1, 0)
Question 6
1 / -0.25

f(x) = x² - x sin x is

A
↑for 0 ≤x ≤p/2

B
↓ for 0 ≤x ≤p/2

C
↓ for [π/4, π/2]

D
None of these
Question 7
1 / -0.25

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

A
1

B
2

C
3

D
None of these
Question 8
1 / -0.25

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

A
f(x) – g(x)

B
f(x) – 2g(x)

C
f(x) + 3g(x)

D
None of these
Question 9
1 / -0.25

Equation 3x² + 4ax + b = 0 has at least one root in (0, 1) if

A
4a + b + 3 = 0

B
2a + b + 1 = 0

C
b = 0, a = ^–

D
None of these
Question 10
1 / -0.25

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

A
f(x) = x³ – x, x Î[0, 1]

B
f(x) =

C
f(x) = e^x , x ∈[–3, 3]

D
f(x) = 1 – , x ∈[–1, 1]
Question 11
1 / -0.25

LMVT is not applicable for which of the following ?

A
f(x) = x² , x ∈[3, 4]

B
f(x) = l n x, x ∈[1, 3]

C
f(x) = 4x² – 5x² + x – 2, x ∈[0, 1]

D
f(x) =