Which of the following statement is true for the function f(x) , … - Vidyadip

MCQ

Which of the following statement is true for the function $f(x)\, = \,\,\,\left[ \begin{array}{l}\sqrt x \,\,\,\,\,\,\,\,\,\,\,\,\,\,x\,\, \ge \,1\\{x^3}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0\, \le x\, \le 1\\\frac{{{x^3}}}{3}\, - \,4x\,\,\,\,\,x\,\, < \,0\end{array} \right.$

A
It is monotonic increasing $\forall \,\,x\,\, \in \,\,R$
B
$f' (x)$ fails to exist for $3$ distinct real values of $x$
✓
$f' (x)$ changes its sign twice as $x$ varies from $(-\infty ,\infty )$
D
function attains its extreme values at $x_1$ & $x_2 $, such that $x_1, x_2 > 0$

✓

Answer

Correct option: C.

$f' (x)$ changes its sign twice as $x$ varies from $(-\infty ,\infty )$

c
COMMENTS: function is inc. in $(-\infty , -2) (0 , \infty )$

function is dec. in $(-2,0)$

$x = -2 \rightarrow$ local maxima

$x = 0 \rightarrow $ local minima

Derivable $\forall \,x\, \in \,R- {0,1} -\left[ \begin{array}{l}f\,'({0^ + })\, = \,0\,\,,\,\,f\,'({0^ - })\, = \, - 4\\f\,'\,({1^ + })\, = \,1/2\,,\,f\,'\,({1^ - })\, = \,3\end{array} \right.$

Continuous $\forall \,x\, \in \,R$.

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