If f(x) = * 20 c e^x + ax, & x < 0 b (x - 1) ^2 , & x 0 . is diff… - Vidyadip

MCQ

If $f(x) = \left\{ {\begin{array}{*{20}{c}}{{e^x} + ax,}&{x < 0}\\{b{{(x - 1)}^2},}&{x \ge 0}\end{array}} \right.$ is differentiable at $x = 0,$ then $(a,\,b)$ is

A
$( - 3,\, - 1)$
✓
$( - 3,\,\,1)$
C
$(3,\,\,1)$
D
$(3,\, - 1)$

✓

Answer

Correct option: B.

$( - 3,\,\,1)$

(b)
Given $f(x)$ is differentiable at $x = 0$.Hence, $f(x)$ will be continuous at $x = 0$.
$\mathop {{\rm{lim}}}\limits_{x \to {0^ - }} ({e^x} + ax) = \mathop {{\rm{lim}}}\limits_{x \to {0^ + }} b{(x - 1)^2}$
$\Rightarrow {e^0} + a \times 0 = b{(0 - 1)^2}$
$\Rightarrow b = 1$….. $(i)$
But $f(x)$ is differentiable at $x = 0$, then
$Lf'(x) = Rf'(x)$
$\Rightarrow \frac{d}{{dx}}({e^x} + ax) = \frac{d}{{dx}}b{(x - 1)^2}$
$\Rightarrow {e^x} + a = 2b(x - 1)$

At $x = 0,$
$\Rightarrow {e^0} + a = - 2b$
$\Rightarrow a + 1 = - 2b$
$\Rightarrow a = - 3$
$\Rightarrow (a,\,\,b) = ( - 3,\,\,1)$.

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

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

Solution of the equation $\sqrt {(x + 10)} + \sqrt {(x - 2)} = 6$ are

Let $I_{n}(x)=\int_{0}^{x} \frac{1}{\left(t^{2}+5\right)^{n}} d t, n=1,2,3, \ldots .$ Then

The points $(1, 3)$ and $(5, 1)$ are the opposite vertices of a rectangle. The other two vertices lie on the line $y = 2x + c,$ then the value of c will be

The coefficient of $x^9$ in the polynomial given by $\sum\limits_{r - 1}^{11} {(x + r)\,(x + r + 1)\,(x + r + 2)...\,(x + r + 9)}$ is

Let $L$ be a normal to the parabola $y^2=4 x$. If $L$ passes through the point $(9,6)$, then $L$ is given by

$(A)$ $y-x+3=0$ $(B)$ $y+3 x-33=0$ $(C)$ $y+x-15=0$ $(D)$ $y-2 x+12=0$

$\int_{}^{} {\frac{1}{{{{({e^x} + {e^{ - x}})}^2}}}\;dx = } $

The opposite angular points of a square are $(3,\;4)$ and $(1,\; - \;1)$. Then the co-ordinates of other two points are

If $f(x) = \frac{{2x + 1}}{{3x - 2}}$, then $(fof)(2)$ is equal to

Let $\quad P_1=I=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{array}\right], \quad P_2=\left[\begin{array}{lll}1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0\end{array}\right], \quad P_3=\left[\begin{array}{lll}0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1\end{array}\right], \quad P_4=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0\end{array}\right], \quad P_5=\left[\begin{array}{lll}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0\end{array}\right]$,$P_6=\left[\begin{array}{lll}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0\end{array}\right]$ and $X=\sum_{k=1}^6 P_k\left[\begin{array}{lll}2 & 1 & 3 \\ 1 & 0 & 2 \\ 3 & 2 & 1\end{array}\right] P_k^{\top}$
where $P _{ K }^{ T }$ denotes the transpose of the matrix $P _{ K }$. Then which of the following options is/are correct?
$(1)$ $X -30 I$ is an invertible matrix
$(2)$ The sum of diagonal entries of $X$ is $18$
$(3)$ If $X \left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]=\alpha\left[\begin{array}{l}1 \\ 1 \\ 1\end{array}\right]$, then $\alpha=30$
$(4)$ $X$ is a symmetric matrix

In a box, there are $20$ cards, out of which $10$ are lebelled as $\mathrm{A}$ and the remaining $10$ are labelled as $B$. Cards are drawn at random, one after the other and with replacement, till a second $A-$card is obtained. The probability that the second $A-$card appears before the third $B-$card is