1 3x 3

1 3x 3 DEFAULT

Find the stationary points of the curve y (x)= 1/3x^3 - 5/2x^2 + 4x and classify them.

To find the stationary points of the curve y(x), you must first differentiate the equation for y(x) in terms of x. This gives d(y(x))/dx = x^2 -5x +4. Now set this differential equal to zero and solve for x (as at a stationary point the gradient of a curve is equal to zero), to find the x coordinates of the stationary points. This quadratic can be factorised to give (x-4)(x-1) = 0, so x is equal to 4 and 1 for the two stationary points respectively. To find the y coordinate of the first stationary point when x=1, simply put this value into the equation for y(x). This gives a stationary point of (1,11/6). For the stationary point when x=4, do the same to obtain the y coordinate. This gives a stationary point (4,-8/3).To classify the stationary points, the second order differential for y(x) must be found, which is d^2(y(x))/dx^2 = 2x -5. Now for the two stationary points, substitute into the second order differential the two x coordinates. For x=1, d^2(y(x))/dx^2 = -3, which is less than zero so (1,11/6) is a local maximum. When x=4, d^2(y(x))/dx^2 = 3, which is greater than zero meaning (4,-8/3) is a local minimum.

Sours: https://www.mytutor.co.uk/answers/31521/A-Level/Maths/Find-the-stationary-points-of-the-curve-y-x-1-3x-3-5-2x-2-4x-and-classify-them/

Linear equations with one unknown

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     1/3*x-3-(9)=0 

Step by step solution :

Step  1  :

1 Simplify — 3

Equation at the end of step  1  :

1 ((— • x) - 3) - 9 = 0 3

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  3  as the denominator :

3 3 • 3 3 = — = ————— 1 3

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

x - (3 • 3) x - 9 ——————————— = ————— 3 3

Equation at the end of step  2  :

(x - 9) ——————— - 9 = 0 3

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  3  as the denominator :

9 9 • 3 9 = — = ————— 1 3

Adding fractions that have a common denominator :

 3.2       Adding up the two equivalent fractions

(x-9) - (9 • 3) x - 36 ——————————————— = —————— 3 3

Equation at the end of step  3  :

x - 36 —————— = 0 3

Step  4  :

When a fraction equals zero :

 4.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

x-36 ———— • 3 = 0 • 3 3

Now, on the left hand side, the  3  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   x-36  = 0

Solving a Single Variable Equation :

 4.2      Solve  :    x-36 = 0 

 Add  36  to both sides of the equation : 
                      x = 36

One solution was found :

                   x = 36
Sours: https://www.tiger-algebra.com/drill/1/3x-3=9/
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Properties of a straight line

Rearrange:

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

                     y-(1/3*x-3)=0 

Step  1  :

1 Simplify — 3

Equation at the end of step  1  :

1 y - ((— • x) - 3) = 0 3

Step  2  :

Rewriting the whole as an Equivalent Fraction :

 2.1   Subtracting a whole from a fraction

Rewrite the whole as a fraction using  3  as the denominator :

3 3 • 3 3 = — = ————— 1 3

Equivalent fraction : The fraction thus generated looks different but has the same value as the whole

Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator

Adding fractions that have a common denominator :

 2.2       Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator

Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

x - (3 • 3) x - 9 ——————————— = ————— 3 3

Equation at the end of step  2  :

(x - 9) y - ——————— = 0 3

Step  3  :

Rewriting the whole as an Equivalent Fraction :

 3.1   Subtracting a fraction from a whole

Rewrite the whole as a fraction using  3  as the denominator :

y y • 3 y = — = ————— 1 3

Adding fractions that have a common denominator :

 3.2       Adding up the two equivalent fractions

y • 3 - ((x-9)) 3y - x + 9 ——————————————— = —————————— 3 3

Equation at the end of step  3  :

3y - x + 9 —————————— = 0 3

Step  4  :

When a fraction equals zero :

 4.1    When a fraction equals zero ...

Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.

Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.

Here's how:

3y-x+9 —————— • 3 = 0 • 3 3

Now, on the left hand side, the  3  cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape :
   3y-x+9  = 0

Equation of a Straight Line

 4.2     Solve   3y-x+9  = 0

Tiger recognizes that we have here an equation of a straight line. Such an equation is usually written y=mx+b ("y=mx+c" in the UK).

"y=mx+b" is the formula of a straight line drawn on Cartesian coordinate system in which "y" is the vertical axis and "x" the horizontal axis.

In this formula :

y tells us how far up the line goes
x tells us how far along
m is the Slope or Gradient i.e. how steep the line is
b is the Y-intercept i.e. where the line crosses the Y axis

The X and Y intercepts and the Slope are called the line properties. We shall now graph the line  3y-x+9  = 0 and calculate its properties

Graph of a Straight Line :

Calculate the Y-Intercept :

Notice that when x = 0 the value of y is -3/1 so this line "cuts" the y axis at y=-3.00000

  y-intercept = -9/3 = -3

Calculate the X-Intercept :

When y = 0 the value of x is 9/1 Our line therefore "cuts" the x axis at x= 9.00000

  x-intercept = 9/1 = 9.00000

Calculate the Slope :

Slope is defined as the change in y divided by the change in x. We note that for x=0, the value of y is -3.000 and for x=2.000, the value of y is -2.333. So, for a change of 2.000 in x (The change in x is sometimes referred to as "RUN") we get a change of -2.333 - (-3.000) = 0.667 in y. (The change in y is sometimes referred to as "RISE" and the Slope is m = RISE / RUN)

  Slope = 0.667/2.000 = 0.333

Geometric figure: Straight Line

  1.   Slope = 0.667/2.000 = 0.333
  2.   x-intercept = 9/1 = 9.00000
  3.   y-intercept = -9/3 = -3
Sours: https://www.tiger-algebra.com/drill/y=1/3x-3/

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