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Nostrana [21]
2 years ago
9

Find the maximum and minimum values of the curve y=2x³-3x²-12x+10​

Mathematics
1 answer:
Ray Of Light [21]2 years ago
5 0

\underline{  \orange{\huge \boxed{ \frak{Answer : }}}}

Let ,

\sf \large \color{purple} y = 2 {x}^{3}   - 3 {x}^{2}  - 12x + 10  \: --( \: 1 \: )

\: \: \:

Now , Diff wrt ' x ' , we get :

\sf \:  \frac{dy}{dx}  =  \frac{d}{dx} (2 {x}^{3}  - 3 {x}^{2}  - 12x + 10) \\  \sf \: \sf \:  \frac{dy}{dx}  =  \frac{d}{dx} \: 2(3 {x}^{2} ) -  \frac{d}{dx} 3 {x}^{2}  -  \frac{d}{dx} 12x +  \frac{d}{dx} 10 \\  \sf \: \frac{dy}{dx}  =2(3 {x}^{2} ) - 3(2x) - 12(1) + 0 \\  \sf \: \frac{dy}{dx}  =6 {x}^{2}  - 6x - 12 + 0 \\  \: \sf \red{\frac{dy}{dx}  = 6 {x}^{2} 6x - 12 -- (2)}

\: \: \:

For maxima or minima \frac{dy}{dx} = 0

\: \: \:

\sf \: 6 {x}^{2}  - 6x - 12 = 0

\: \: \:

Divided by 6 on both side , we get.

\: \: \:

\sf \:  {x}^{2}  - x - 2 = 0 \\  \sf \:  {x}^{2}  - 2x + x - 2 = 0 \\  \sf \: x(x - 2) + 1(x - 2) = 0 \\  \sf \: (x - 2)(x + 1) = 0 \\  \sf \: x - 2 = 0 \:  \:  \bold or \:  \: x + 1 = 0 \\ \sf \fbox{x = 2 \: }  \: \bold or \:  \fbox{ x =  - 1}

\: \: \:

Again Diff wrt ‘ x ’ , we get.

\sf \: \frac{d}{dx}  =(\frac{dy}{dx}  ) = 6\frac{d}{dx}  - 6\frac{d}{dx}x - \frac{d}{dx}12 \\  \sf \:  \frac{ {d}^{2}y }{ {dx}^{2} }  = 6(2x) - 6(1) - 0 \\ \sf \: \sf \bold{  \frac{ {d}^{2}y }{ {dx}^{2} }  =12x - 6}

\: \: \:

At x = 2

\: \: \:

\sf \:  \frac{ {d}^{2}y }{ {dx}^{2} }  =12(2) - 6 \\  \:  \:  \:  \sf \:  = 24 - 6 \\   \:  \:  \:  \: \sf \red{  = 18  > 0}

At x = -1

\: \: \:

\sf \:  \frac{ {d}^{2}y }{ {dx}^{2} }  =12( - 1) - 6  \\   \:  \:  \: \sf \:   = - 12 - 6 \\  \:  \:  \:  \: \sf \red{  =  - 18 < 0 }

\: \: \:

x = 2 gives minima value of function.

\: \: \:

x = -1 gives maxima value of function.

\: \: \:

Now, put x = 2 in eqⁿ ( 1 )

\: \: \:

\sf \: y \: minima  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \: = 2( {2})^{3}  - 3 ({2})^{2}  - 12(2) + 10 \\  \:  \:  \:   \:  \:  \: \:  \:  \:  \:  \:  \:  \: \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:    \:  \:  \:   \sf \:   \:  \:   \:  = 2(8) - 3(4) - 24 + 10 \\ \:  \:  \: \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:    \sf \:   \:  \:  \:  \:  = 16  - 12 - 24 + 10 \\\sf \:   \:  \:  \:  \:  \:  \:  \:  \:  \: =  - 20 + 10 \\\sf \color{red}{\boxed{  =  - 10}}

\: \: \:

<u>The </u><u>Point </u><u>of </u><u>minima </u><u>is </u><u>(</u><u> </u><u>2</u><u> </u><u>,</u><u> </u><u>-</u><u>1</u><u>0</u><u> </u><u>)</u><u>.</u>

\: \: \:

Now , put x = -1 in eqⁿ ( 1 )

\sf \: y \: maxima  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:  \:   = 2( { - 1})^{3}  - 3 ({ - 1})^{2}  - 12( - 1) + 10  \\\sf  \color{red}{\boxed{  =  17}}

\: \: \:

<u>The point of maxima value is ( -1 , 17 )</u><u>.</u>

\: \: \:

<u>\:  \:</u>

Hope Helps! :)

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