Question

a curve has the equation y=x^3+8x^2+5x.Work out the coordinates of the two Turing points

Answer 1

$\huge\underline{\red{Given.\:\:...}}$

a curve has the equation y=x^3+8x^2+5x.Work out the coordinates of the two Turing points

$\huge\underline{\red{Answer.\:\:...}}$

(-5,56) and (-1/3,-0.815)

$\huge\underline{\red{Explanation.\:\:...}}$

The equation is:

$y = {x}^{3} + {8x}^{3} + 5x$

Now the Turing points will occur at where dy/dx = 0.

Thus:

$\frac{dy}{dx} = {3x}^{2} + 16x + 5$

At dy/dx = 0, we have;

${3x}^{2} + 16x + 5 = 0$

From quadratic formula, we can get;

x = -5 or -1/3

Thus, let's find Yat this points

$y \: = \: { - 5}^{3} + 8( - 5) {}^{2} + 5( - 5) \\ y \: = \: - 125 + 200 - 25 \\ y \: = \: 50$

Also, at x = -1/3 we have;

$y \: = \: ( \frac{ - 1}{3} ) {}^{3} + 8( \frac{ - 1}{3} ) {}^{2} + 5( \frac{ - 1}{3} ) \\ y \: = \: - 0.815$

Thus, the coordinates of the Turing points are (-5,50) and (-1/3,-0.815)

Answer 2

Given:

y=x³+8x²+5x

The Turing points will occur at where dy /dx = 0

Here we have

$\frac{d y}{d x}=3 x^{2}+16 x+5$

put dy/dx = 0

x³+8x²+5x=0

From quadratic equation formula we get two roots

Two roots are x = -5 or x = -1/3

Now put the value of x in given equation

For at point x=-5

y= x ³+ 8x² + 5x

y=(-5)³ + 8(-5)² + 5(-5)

y=-125 + 8(25) - 25

y= -125 + 200 - 25

y = 50

For at point x= -1/3

$\begin{array}{l}y=\left(\frac{-1}{3}\right)^{3}+8\left(\frac{-1}{3}\right)^{2}+5\left(\frac{-1}{3}\right) \\y=-0.815\end{array}$

Hence the  two Turing points are (-5 , 50) and (-1/3 , -0.815)