If $f(x) = \left\{ \begin{array}{r} ax^{2} + b,\ \ b
eq 0,\ x \leq 1 \ x^{2}b + ax + c,\ \ \ x 1 \end{array} ight.\$, then, $f(x)$ is continuous and differentiable at $x = 1$, if

Question

If $f(x) = \left\{ \begin{array}{r} ax^{2} + b,\ \ b 
eq 0,\ x \leq 1 \ x^{2}b + ax + c,\ \ \ x > 1 \end{array} ight.\$, then, $f(x)$ is continuous and differentiable at $x = 1$, if

Dr. Abubacker · Accepted Answer

(a)
Here, $f^{'}(x) = \left\{ \begin{array}{r}
2ax,\ \ b 
eq 0,\ x \leq 1 \
2bx + a,\ \ x > 1
\end{array} ight.\$
Since, $f(X)$ is continuous at $x = 1$
$	herefore\ \ \ \lim_{h ightarrow 0}\ \ f(x) = \lim_{h ightarrow 1^{+}}\ f(x)$
$\Rightarrow \ \ \ a + b = b + a + c\  \Rightarrow c = 0$
Also, $f(x)$ is differentiable at $x = 1$
$	herefore\$(LHD at $x = 1$)=(RHD at $x = 1$)
$\Rightarrow \ \ 2a = 2b(1) + a\ \  \Rightarrow \ \ a = 2b$

Q. If $f(x) = \left\{ \begin{array}{r} ax^{2} + b,\ \ b \neq 0,\ x \leq 1 \\ x^{2}b + ax + c,\ \ \ x > 1 \end{array} \right.\$, then, $f(x)$ is continuous and differentiable at $x = 1$, if

Explanation

Reviewed by Dr. Abubacker

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