The function f(x) = * 20 c e^ 2x - 1 &,& x 0 ax + b x^2 2 - 1 &,&… - Vidyadip

MCQ

The function $f(x) = \left\{ {\begin{array}{*{20}{c}}{{e^{2x}} - 1}&,&{x \le 0}\\{ax + \frac{{b{x^2}}}{2} - 1}&,&{x > 0}\end{array}} \right.$ is continuous and differentiable for

A
$a = 1,\,b = 2$
B
$a = 2,\,b = 4$
✓
$a = 2,\,$ any $b$
D
Any $a,\,\,\,b = 4$

✓

Answer

Correct option: C.

$a = 2,\,$ any $b$

(c)
$f$ is continuous at $x = 0$,
$f({0^ - }) = f({0^ + }) = f(0) = - 1$
Also $Lf'(0) = Rf'(0)$
$\Rightarrow \mathop {\lim }\limits_{h \to 0} \frac{{f(0 - h) - f(0)}}{{ - h}} = \mathop {\lim }\limits_{h \to 0} \frac{{f(0 + h) - f(0)}}{h}$
$\Rightarrow \mathop {\lim }\limits_{h \to 0} \left( {\frac{{{e^{ - 2h}} - 1 + 1}}{{ - h}}} \right) = \mathop {\lim }\limits_{h \to 0} \left( {\frac{{ah + \frac{{b{h^2}}}{2} - 1 + 1}}{h}} \right)$
$\Rightarrow \mathop {\lim }\limits_{h \to 0} \left( {\frac{{ - 2{e^{ - 2h}}}}{{ - 1}}} \right) = \mathop {\lim }\limits_{h \to 0} \left( {a + \frac{{bh}}{2}} \right)$
$\Rightarrow 2 = a + 0$
$\Rightarrow a = 2,\,\,b$ any number.

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