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Question Number 26381 by Tinkutara last updated on 24/Dec/17

Commented by Tinkutara last updated on 24/Dec/17

Should answer be (4) or not?

Commented by mrW1 last updated on 24/Dec/17

$$Ithink1and4aretrue.$$

Commented by Tinkutara last updated on 24/Dec/17

But answer given is only (1).

Commented by prakash jain last updated on 24/Dec/17

$$y^{\prime }=\frac{dy}{dx}$$$$\frac{d^{2}y}{{dx}^2}=\underset{h\to 0^{+}}{\lim }\frac{y^{\prime }(x+h)-y^{\prime }}{h}\left(I\right)$$$$\frac{d^{2}y}{{dx}^2}=\underset{h\to 0^{+}}{\lim }\frac{y^{\prime }\left(x\right)-y^{\prime }(x-h)}{h}\left(II\right)$$$$if{y}^{\prime }isincreasing\frac{d^{2}y}{{dx}^2}>0$$

Commented by mrW1 last updated on 25/Dec/17

$${let}^{\prime }slookatanexample:$$$$y^{\prime }={(x-3)}^3+2$$$$itisincreasingforallx.$$$$but{y}^{″}=3{(x-3)}^2⩾0not>0$$$$thatmeansif{y}^{\prime }isincreasing,$$$$\frac{d^{2}y}{{dx}^2}couldalsobeequaltozero.$$

Commented by mrW1 last updated on 25/Dec/17

$$sotheanswerisindeedonly\left(1\right),$$$$sinceatx=0,\frac{d^{2}y}{{dx}^2}couldbe=0.$$$$forexampleifthecurveis$$$$y=x^{n}withn⩾3.$$

Commented by prakash jain last updated on 25/Dec/17

$$\mathrm{correct}.\mathrm{For}\mathrm{strictly}\mathrm{increasing}$$$$\mathrm{function}\mathrm{derivate}\mathrm{is}\mathrm{greater}\mathrm{than}$$$$\mathrm{or}\mathrm{equal}\mathrm{to}0.$$

Commented by mrW1 last updated on 25/Dec/17

$$thanksforcheckingsir!youmade$$$$andmakeagreatcontributiontothis$$$$community.$$

Commented by Tinkutara last updated on 25/Dec/17

Thanks to both mrW1 and prakash jain!

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