The values of p\; and\; q for which the function

 f(x)=\left\{\begin{matrix} \frac{\sin (p+1)x+\sin x}{x} &,x< 0 \\ q&,x=0 \\ \frac{\frac{q}{\sqrt{x+x^{2}-\sqrt{x}}}}{x^{3/2}}&,x> 0 \end{matrix}\right.

is continuous for all x in R,are

 

  • Option 1)

    p=-\frac{3}{2},q=\frac{1}{2}\;

  • Option 2)

    \; \; p=\frac{1}{2},q=\frac{3}{2}\;

  • Option 3)

    \; p=\frac{1}{2},q=-\frac{3}{2}\;

  • Option 4)

    \; \; p=\frac{5}{2},q=\frac{1}{2}

 

Answers (1)

As we learnt in

Continuity -

If the function is continuous, Its graph does not break but for discontinuous functions there is a break in the graph.

- wherein

 

 f(x)\:=\left\{\begin{matrix} \frac{sin(p+1)x+sinx}{x} & x< 0\\ q & =0\\ \frac{\sqrt{x+x^{2}}-\sqrt{x}}{x^\frac{3}{2}} & x> 0 \end{matrix}\right.

f(x)\:=\left\{\begin{matrix} \frac{2sin\frac{(p+2)x}{2}.cos\frac{px}{2}}{x} & x< 0\\ q & x=0\\ \frac{(x+x)^{2}-x}{x.x^\frac{1}{2}}.\frac{1}{\sqrt{x+x^{2}}+\sqrt{x}} & x> 0 \end{matrix}\right.

f(x)=\left\{\begin{matrix} \frac{2sin\frac{(p+2)x}{2}}{\frac{(p+2)x}{2}\times2}(p+2).cos\frac{px}{2}& x< 0\\ q & x=0\\ \frac{x^{2}}{x.x^\frac{1}{2}.\sqrt{x}(1+\sqrt{1+x})} & x> 0 \end{matrix}\right.

Since f(x) is continuous so that

=> P+2 = q = \frac{1}{2}

\therefore P = -\frac{3}{2}

q = \frac{1}{2}


Option 1)

p=-\frac{3}{2},q=\frac{1}{2}\;

This solution is correct.

Option 2)

\; \; p=\frac{1}{2},q=\frac{3}{2}\;

This solution is incorrect 

Option 3)

\; p=\frac{1}{2},q=-\frac{3}{2}\;

This solution is incorrect 

Option 4)

\; \; p=\frac{5}{2},q=\frac{1}{2}

This solution is incorrect 

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