Purchase Solution

Stationary point

Not what you're looking for?

Ask Custom Question

This question concerns the function f(x)=x^3 +3x^2 −24x+40.
(a) Find the stationary points of this function.
(b) (i) Using the strategy to apply the First Derivative Test, classify the left-hand stationary point found in part (a).
(ii) Using the Second Derivative Test, classify the right-hand stationary point found in part (a).
(c) Find the y-coordinate of each of the stationary points on the graph of the function f (x), and also evaluate f (0).
(d) Hence draw a rough sketch of the graph of the function f(x).

Purchase this Solution
Solution Summary

For given function, stationary point is obtained. Followed by using first and second derivatives, left hand and right hand are classified. Finally y co-ordinate of the stationary point is obtained and the functions is drawn.

Solution Preview

(a)
f(x) = x^3 +3x^2 −24x+40

df/dx = 3x^2 + 6x - 24

For stationary points,
df/dx = 0
=> 3x^2 + 6x - 24 = 0
=> x^2 + 2x - 8 = 0
=> x^2 + 4x - 2x - 8 = 0
=> x (x + 4) - 2 (x + 4) = 0
=> (x+4) (x - 2) = 0
=> x = -4, 2
Hence, stationary points are: -4, 2

(b)(i)
First Derivative test: Find the slopes of the function near the stationary points (both the sides).

Left side of x = -4, x = -4-h, where h is a positive small fractional (<1) number
df/dx|(x : -4 -h) = 3(-4-h)^2 + 6(-4-h) - 24
= 3(16 ...

Solution provided by:
Amrit Lal Ahuja, PhD (IP)
Education
  • BEng, Allahabad University, India
  • MSc , Pune University, India
  • PhD (IP), Pune University, India
Recent Feedback
  • " In question 2, you incorrectly add in the $3.00 dividend that was just paid to determine the value of the stock price using the dividend discount model. In question 4 response, it should have also been recognized that dividend discount models are not useful if any of the parameters used in the model are inaccurate. "
  • "feedback: fail to recognize the operating cash flow will not begin until the end of year 3."
  • "Answer was correct"
  • "Great thanks"
  • "Perfect solution..thank you"
Purchase this Solution
Free BrainMass Quizzes
Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

Solving quadratic inequalities

This quiz test you on how well you are familiar with solving quadratic inequalities.

Probability Quiz

Some questions on probability

Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

View More Free Quizzes
Related BrainMass Solutions

  • Evaluating Stationary Points

    Thus the left handed stationary point is a local maximum. Alternatively, f"(x)=12x+6, when x=-3, f"(-3)=-36+6<0, we could have local maximum at the left handed stationary point (-3,88).

  • find out the stationary point, saddle point, min and max

    The solution provides detailed explanation how to find out the stationary point, saddle point, min and max.

  • Find Stationary Points and Inflexions of a cubic polynomial

    (c) Find the coordinates of the point of inflexion,that is find the x and y values of the point of inflexion. (d) Do a specific example with numbers substituted for a,b,c and where two stationary points occur.

  • Stationary Points on a Cubic Curve

    The relative Maximum point is (2, f(2)) = (2, 43) The relative Minimum point is (3, f(3)) = (3, 42.5). The stationary points on a cubic curve is determined. Neat and step-by-step solution is provided.

  • Curve Sketching, Integration, Stationary Points and Asymptotes

    stationary point at x can be determined by examining the second derivative f''(x): • If f''(x) < 0, the stationary point at x is a maximal extremum

  • Finding the Critical Points for Functions of Two Variables

    hx = 2(x2 - 4)*2x = 4x(x2 - 4) = 4x3 - 16x = 0 implies that x = 0, 2, -2 (stationary points) hy = 2y = 0 implies that y = 0 (stationary point) Therefore, the stationary points are (0, 0), (2, 0), (-2, 0) Now, hxx = 12x2 - 16, hyy = 2, and hxy = 0

  • Function, derivatives, error, and stationary points.

    Locate and classify all the stationary points of f. (a) We have the following derivative calculations. (b) From (a), we can get the second order Taylor polynomial for near the point .

  • Water Flow

    Determine the velocity with which water exits the hole when the cylinder is filled to a point 4.9 meters above the hole. Do not assume that the water in the cylinder is stationary.

  • Stationary Points and Minimum Surface Area

    c). z = 3x - 6 y = 0 so y = ½ *x ; z = 3 y - 6x = 0 so x = ½ * y ; So x = ½ * y = ½ * (1/4) *x = 1/8 * x so 1 = 1/8 * x x = 2, so y = 2 So the stationary point is (2, 2). b).

View More Related Solutions