Question

P(x)= -0.0013x^3+0.3x^2+8x-372. Find the maximum without graphing. Please show step by step.

Answer 1

The Maximum value is
`P(x)=2076.227`

Explanation:

Given,

`P(x)=-0.0013* x^(3) +0.3* x^(2) +8x-372` (equation-1)

Differentiate above equation with respect to 'x',

`P'(x)=-0.0039* x^(2) +0.6x+8` --- (equation 2)

Again differentiate above equation with respect to 'x',

`P''(x)=-0.0078* x +0.6` ------- (equation 3)

From equation-2 we see,

The value of
`a=-0.0039` ,
`b=0.6` ,
`c=8`.

Now, for maximum or minimum, the first derivative must be 0.

For maximum,
`P''(x)<0`

So,
`P'(x)=-0.0039* x^(2) +0.6x+8 = 0`

Using the quadratic formula, we find the roots of
`P'(x)`
`x=\frac{-b\pm \sqrt{b^(2)-4ac } }{2a}`

`x=\frac{-0.6\pm \sqrt{0.6^(2)-4* -0.0039* 8 } }{2* -0.0039}`

`x=(-0.6\pm 0.696)/(-0.0078)`

`x=-12.3` or
`x=166.15`

For
`x=-12.3`,

`P''(x)=(6*-0.0013* -12.3 )+(2* 0.3)`

`P''(x)=0.696>0`

Which is minimum value at
`x=-12.3`

And for
`x=166.15`,

`P''(x)=(6*-0.0013* 166.15 )+(2* 0.3)`

`P''(x)=-0.696< 0`

Which is maximum value at
`x=166.15`

Plug
`x=166.15` in equation-1,

`P(x)=-0.0013* 166.15^(3) +0.3* 166.15^(2) +8* 166.15-372`

`P(x)=2076.227`

So the Maximum value is
`P(x)=2076.227`