Question

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?
Interval Function
(a) $$\left( { - \infty ,\,\infty } \right)$$ $${x^3} - 3{x^2} + 3x + 3$$
(b) $$\left[ {2,\,\infty } \right)$$ $$2{x^3} - 3{x^2} - 12x + 6$$
(c) $$\left( { - \infty ,\,\frac{1}{3}} \right]$$ $$3{x^2} - 2x + 1$$
(d) $$\left( { - \infty ,\, - 4} \right)$$ $${x^3} + 6{x^2} + 6$$

A. a
B. b
C. c  
D. d
Answer :   c
Solution :
Clearly function $$f\left( x \right) = 3{x^2} - 2x + 1$$     is increasing when $$f'\left( x \right) = 6x - 2 \geqslant 0 \Rightarrow x \in \left[ {\frac{1}{3},\infty } \right)$$
$$\therefore f\left( x \right)$$  is incorrectly matched with $$\left( { - \infty ,\frac{1}{3}} \right)$$

Releted MCQ Question on
Calculus >> Function

Releted Question 1

Let $$R$$ be the set of real numbers. If $$f:R \to R$$   is a function defined by $$f\left( x \right) = {x^2},$$   then $$f$$ is:

A. Injective but not surjective
B. Surjective but not injective
C. Bijective
D. None of these.
View Answer
Releted Question 2

The entire graphs of the equation $$y = {x^2} + kx - x + 9$$     is strictly above the $$x$$-axis if and only if

A. $$k < 7$$
B. $$ - 5 < k < 7$$
C. $$k > - 5$$
D. None of these.
View Answer
Releted Question 3

Let $$f\left( x \right) = \left| {x - 1} \right|.$$    Then

A. $$f\left( {{x^2}} \right) = {\left( {f\left( x \right)} \right)^2}$$
B. $$f\left( {x + y} \right) = f\left( x \right) + f\left( y \right)$$
C. $$f\left( {\left| x \right|} \right) = \left| {f\left( x \right)} \right|$$
D. None of these
View Answer
Releted Question 4

If $$x$$ satisfies $$\left| {x - 1} \right| + \left| {x - 2} \right| + \left| {x - 3} \right| \geqslant 6,$$       then

A. $$0 \leqslant x \leqslant 4$$
B. $$x \leqslant - 2\,{\text{or}}\,x \geqslant 4$$
C. $$x \leqslant 0\,{\text{or}}\,x \geqslant 4$$
D. None of these
View Answer

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