Question

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

Answer 1

Answer:

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

Step-by-step explanation:

Given,

$P(x)=-0.0013\times x^{3} +0.3\times x^{2} +8x-372$ (equation-1)

Differentiate above equation with respect to 'x',

$P'(x)=-0.0039\times x^{2} +0.6x+8$ --- (equation 2)

Again differentiate above equation with respect to 'x',

$P''(x)=-0.0078\times x +0.6$ ------- (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, $P'(x)=-0.0039\times x^{2} +0.6x+8 = 0$

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

$x=\frac{-0.6\pm \sqrt{0.6^{2}-4\times -0.0039\times 8 } }{2\times -0.0039}$

$x=\frac{-0.6\pm 0.696}{-0.0078}$

$x=-12.3$ or $x=166.15$

For $x=-12.3$,

$P''(x)=(6\times-0.0013\times -12.3 )+(2\times 0.3)$

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

Which is minimum value at $x=-12.3$

And for $x=166.15$,

$P''(x)=(6\times-0.0013\times 166.15 )+(2\times 0.3)$

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

Which is maximum value at  $x=166.15$

Plug $x=166.15$ in equation-1,

$P(x)=-0.0013\times 166.15^{3} +0.3\times 166.15^{2} +8\times 166.15-372$

$P(x)=2076.227$

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