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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

I think 1 and 4 are true.

$${I}\:{think}\:\mathrm{1}\:{and}\:\mathrm{4}\:{are}\:{true}. \\ $$

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′=(dy/dx) (d^2 y/dx^2 )=lim_(h→0^+ ) ((y′(x+h)−y′)/h) (I) (d^2 y/dx^2 )=lim_(h→0^+ ) ((y′(x)−y′(x−h))/h) (II) if y′ is increasing (d^2 y/dx^2 )>0

$${y}'=\frac{{dy}}{{dx}} \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }=\underset{{h}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{{y}'\left({x}+{h}\right)−{y}'}{{h}}\:\:\left({I}\right) \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }=\underset{{h}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{{y}'\left({x}\right)−{y}'\left({x}−{h}\right)}{{h}}\:\left({II}\right) \\ $$$${if}\:{y}'\:{is}\:{increasing}\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }>\mathrm{0} \\ $$

Commented by mrW1 last updated on 25/Dec/17

let′s look at an example: y′=(x−3)^3 +2 it is increasing for all x. but y′′=3(x−3)^2 ≥0 not >0 that means if y′ is increasing, (d^2 y/dx^2 ) could also be equal to zero.

$${let}'{s}\:{look}\:{at}\:{an}\:{example}: \\ $$$${y}'=\left({x}−\mathrm{3}\right)^{\mathrm{3}} +\mathrm{2} \\ $$$${it}\:{is}\:{increasing}\:{for}\:{all}\:{x}. \\ $$$${but}\:{y}''=\mathrm{3}\left({x}−\mathrm{3}\right)^{\mathrm{2}} \geqslant\mathrm{0}\:{not}\:>\mathrm{0} \\ $$$${that}\:{means}\:{if}\:{y}'\:{is}\:{increasing}, \\ $$$$\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:{could}\:{also}\:{be}\:{equal}\:{to}\:{zero}. \\ $$

Commented by mrW1 last updated on 25/Dec/17

so the answer is indeed only (1), since at x=0, (d^2 y/dx^2 ) could be =0. for example if the curve is y=x^n with n≥3.

$${so}\:{the}\:{answer}\:{is}\:{indeed}\:{only}\:\left(\mathrm{1}\right), \\ $$$${since}\:{at}\:{x}=\mathrm{0},\:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:{could}\:{be}\:=\mathrm{0}. \\ $$$${for}\:{example}\:{if}\:{the}\:{curve}\:{is} \\ $$$${y}={x}^{{n}} \:{with}\:{n}\geqslant\mathrm{3}. \\ $$

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

correct. For strictly increasing function derivate is greater than or equal to 0.

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

Commented by mrW1 last updated on 25/Dec/17

thanks for checking sir! you made and make a great contribution to this community.

$${thanks}\:{for}\:{checking}\:{sir}!\:{you}\:{made} \\ $$$${and}\:{make}\:{a}\:{great}\:{contribution}\:{to}\:{this}\: \\ $$$${community}. \\ $$

Commented by Tinkutara last updated on 25/Dec/17

Thanks to both mrW1 and prakash jain!

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