Question

Let $f\left(x\right)=\varphi \left(x\right)+\psi \left(x\right)$ and ${\varphi }^{{}^{\prime }}\left(a\right)$, ${\psi }^{{}^{\prime }}\left(a\right)$ are finite and definite. Then

• A. $f\left(x\right)$ is continuous at $x=a$
• B. $f\left(x\right)$ is differentiable at $x=a$
• C. $f^{{}^{\prime }}\left(x\right)$ is continuous at $x=a$
• D. $f^{{}^{\prime }}\left(x\right)$ is differentiable at $x=a$

Answer 1

Answer: (A), (B)

The correct options are
A $f\left(x\right)$ is continuous at $x=a$
B $f\left(x\right)$ is differentiable at $x=a$

$f\left(x\right)=\psi \left(x\right)+\varphi \left(x\right)Given{\varphi }^{\prime }\left(a\right)and{\psi }^{\prime }\left(a\right)aredefiniteandfinite\Rightarrow f^{\prime }\left(a\right)isdefiniteandfinite\Rightarrow f\left(a\right)iscontinuousanddifferentiable$
Hence, options 'A' and 'B' are correct.