Question

If $f\left(x\right)=p|sinx|+q{e}^{|x|}+r|x{|}^3$ and if f(x) is differentiable at x = 0, then :

• A. $p,q,rϵR$
• B. p =0; q=0; r is any real number
• C. q = 0; r=0; p is any real number
• D. r = 0: p = 0; q is any real number.

Answer 1

The correct option is B p =0; q=0; r is any real number

$f\left(x\right)=p|sinx|+q{e}^{|x|}+r|x{|}^3$
$R{f}^{\prime }\left(0\right)=\underset{h\to 0}{lim}\frac{(psinh+q{e}^h+r{h}^3)-\left(q\right)}{h}$
$=\underset{h\to 0}{lim}\frac{psinh+r{h}^3}{h}+\frac{q(e^h-1)}{h}$
$=p+q$
$L{f}^{\prime }\left(0\right)=\underset{h\to 0}{lim}\frac{+psinh+q{e}^{+h}+r{h}^3-q}{-h}$
$=\underset{h\to 0}{lim}-p\frac{sinh}{h}-r{h}^2+\frac{q(e^{+h}-1)}{-h}$
$=-p-q$.
f(x) is differentiable at x = 0
$\Rightarrow p+q=-p-q$
$\Rightarrow p+q=0$ which is satisfied by choice (b)