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
(c) $f$ is continuous at $x = 0$, $f({0^ - }) = f({0^ + }) = f(0) = - 1$

Also $Lf'(0) = Rf'(0)$

==> $\mathop {\lim }\limits_{h \to 0} \frac{{f(0 - h) - f(0)}}{{ - h}} = \mathop {\lim }\limits_{h \to 0} \frac{{f(0 + h) - f(0)}}{h}$

==> $\mathop {\lim }\limits_{h \to 0} \left( {\frac{{{e^{ - 2h}} - 1 + 1}}{{ - h}}} \right) = \mathop {\lim }\limits_{h \to 0} \left( {\frac{{ah + \frac{{b{h^2}}}{2} - 1 + 1}}{h}} \right)$

==> $\mathop {\lim }\limits_{h \to 0} \left( {\frac{{ - 2{e^{ - 2h}}}}{{ - 1}}} \right) = \mathop {\lim }\limits_{h \to 0} \left( {a + \frac{{bh}}{2}} \right)$

==> $2 = a + 0$ ==> $a = 2,\,\,b$ any number.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "M.C.Q (1 Marks)" from 5. continuity and differentiation→5. continuity and differentiation — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

The value of $\int\frac{\text{d}(\text{x}^2+1)}{\sqrt{\text{x}^2+2}}$ is:

$2\sqrt{\text{x}^2+2}+\text{c}$
$\sqrt{\text{x}^2+2}+\text{c}$
$\text{x}\sqrt{\text{x}^2+2}+\text{c}$
${4}\sqrt{\text{x}^2+2}+\text{c}$

Choose the correct answer from the given four options.
The probability of guessing correctly at least 8 out of 10 answers on a true-false type examination is:

$\frac{7}{64}$
$\frac{7}{128}$
$\frac{45}{1024}$
$\frac{7}{41}$

If $\text{f(x)}=\begin{cases}\frac{\sin(\cos\text{x})-\cos\text{x}}{(\pi-2\text{x})^2},&\text{x}\neq\frac{\pi}{2}\\\text{k},&\text{x}=\frac{\pi}{2}\end{cases}$ is continuous at $\text{x}=\frac{\pi}{2},$ then k is equal to:

0
$\frac{1}{2}$
1
-1

Linear programming model which involves funds allocation of limited investment is classified as:

Ordination budgeting model
Capital budgeting models
Funds investment models
Funds origin models.

Let $A=\{1,2,3,5,8,9\}$. Then the number of possible functions $f : A \rightarrow A$ such that $f(m \cdot n)=f(m) \cdot f(n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to $...............$.

Let $N$ denote the set of all natural numbers and $R$ be the relation on $N \times N$ defined by $(a, b)$ $R$ $(c, d)$ if $ad(b + c) = bc(a + d),$ then $R$ is

The area of triangle whose vertices are $(2,-6),(5,4)$ and $(k, 4)$ is 35 sq. unit then, value of $k$ is __________ .

$\int_{\,0}^{\,2a} {f(x)dx = } $

The value of the integral $\int\limits_4^{10} {\frac{{\left[ {{x^2}} \right]dx}}{{\left[ {{x^2} - 28x + 196} \right] + \left[ {{x^2}} \right]}}}$ , where $\left[ x \right]$ denotes the greatest integer less than or equal to $x$, is

If A = $\begin{bmatrix}\alpha&\beta\\ \gamma&-\alpha\end{bmatrix}$is such that A² = I, then:

$1 + \alpha^2 + \beta\gamma = 0$
$1 - \alpha^2 + \beta\gamma = 0$
$1 - \alpha^2 - \beta\gamma = 0$
$1 + \alpha^2 - \beta\gamma = 0$