The function f defined by

\small f(x)=x^{3}-3x^{2}+5x+7 is:

 

  • Option 1)

     increasing in R.

  • Option 2)

     decreasing in R.

  • Option 3)

     decreasing in (0, ∞) and increasing in (−∞, 0).

  • Option 4)

    increasing in (0, ∞) and decreasing in (−∞, 0).

     

 

Answers (1)

As we learnt in 

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 for (a,b)

 

 f\left ( x \right )= x^{3}-3x^{2}+5x+7

f'\left ( x \right )= 3x^{2}-6x+5

D=\left ( 6 \right )^{2}-4\cdot 3\cdot 5

= 36-60< 0

So, 

f'\left ( x \right )> 0     for  x\epsilon R

Correct option is 1.


Option 1)

 increasing in R.

This option is incorrect.

Option 2)

 decreasing in R.

This option is incorrect.

Option 3)

 decreasing in (0, ∞) and increasing in (−∞, 0).

This option is correct.

Option 4)

increasing in (0, ∞) and decreasing in (−∞, 0).

 

This option is incorrect.

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