Let f :R R be a function defined as f ( x ) = l 5, , , , , , , , … - Vidyadip

MCQ

Let $f :R \to R$ be a function defined as $f\left( x \right) = \left\{ \begin{array}{l}
5,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,\,\,\,x \le 1\,\,\,\,\,\,\,\\
a + bx,\,\,\,\,if\,\,\,\,\,\,1 < x < 3\\
b + 5x,\,\,\,\,if\,\,\,\,\,\,3 \le x < 5\\
30,\,\,\,\,\,\,\,\,\,\,if\,\,\,\,\,\,\,x \ge 5
\end{array} \right.\,\,\,\,$ Then $f$ is

A
continuous if $a = 5$ and $b = 5$
B
continuous if $a = 5$ and $b = 10$
C
continuous if $a = 0$ and $b = 5$
✓
not continuous for any values of $a$ and $b$

✓

Answer

Correct option: D.

not continuous for any values of $a$ and $b$

d
For $x=1$

$R.H.L=a+b$

$L.H.L=5$

So to be continuous at $x=1$

$a+b=5$ ..........$(i)$

for $x=3$

$R.H.L=b+15$

$L.H.L=a+3b$

$b+15=a+3b$

$a+2b=15$ ........$(ii)$

for $x=5$

$R.H.L=30$

$L.H.L=b+25$

$b+25=30$

$b=5$.

From equation $(ii)$

$a=10$

but $a=10$ and $b=5$ does not satisfied equation $(i)$

So $f(x)$ is discontinuous for $a \in R$ and $b \in R$

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