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  • Confused! kindly explain, - Let where a,b and d are non-zero real constants. Then : - Limit , continuity and differentiability - JEE Main

Let f(x)=\frac{x}{\sqrt{a^{2}+x^{2}}}- \frac{d-x}{\sqrt{b^{2}+(d-x)^{2}}} , x\epsilon R,

where a,b and d are non-zero real constants. Then : 

  • Option 1)

    f is neither increasing nor decreasing function of x

  • Option 2)

    f ' is not a continuous function of x

  • Option 3)

    f is a decreasing function of x

  • Option 4)

    f is an increasing function of x

Answers (1)

best_answer

 

Condition for increasing functions -

For increasing function tangents drawn at any point on it makes an acute slope with positive x-axis.

M_{T}=tan\theta\geq 0

\therefore \:\:\:\frac{dy}{dx}=f'(x)\geq 0\:\:for\:\:x\epsilon (a,b)

- wherein

Where f(x)  is continuous and differentiable for (a,b)

f{}'(x)=\frac{a^{2}}{(a^{2}+x^{2})\frac{3}{2}}+\frac{b^{2}}{(b^{2}+(d-x^{2}))\frac{3}{2}}>0\: \vee\: x\varepsilon R

f{}(x)\: \: is \: \: increasing\: \: function.

 

 

Condition for Decreasing function -

The tangents drawn at any point on it make an obtuse angle with positive direction of  the x-axis.

M_{T}=tan\theta\leq 0

\therefore \:\:\frac{dy}{dx}\leq 0\:\:\:\:\:x\epsilon (a,b)

- wherein

y=f(x) is continuous and differentiable in (a,b)

 

 


Option 1)

f is neither increasing nor decreasing function of x

Option 2)

f ' is not a continuous function of x

Option 3)

f is a decreasing function of x

Option 4)

f is an increasing function of x
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