generate a continuous and differentiable function f(x) with the following properties: f(x) is decreasing at x=−5 f(x) has a local minimum at x=−3 f(x) has a local maximum at x=3

A function is differentiable at , if the function is continuous at . The function that satisfy the given properties is

Given that:

The function decreases at means that:

The local minimum at and local maximum at means that:

or

Equate both equations to 0

or

Multiply both equations to give y'

Open bracket

Collect like terms

Integrate y'

Express as a function

implies that:

Take LCM

Collect like terms

We can then assume the value of c to be

or any other value less than 3.33

Substitute in

See attachment for the function of f(x)

Read more about continuous and differentiable function at:

link

see details in graph and below

Step-by-step explanation:

There are many ways to generate the function.

We'll generate a function whose first derivative f'(x) satisfies the required conditions, say, a quadratic.

1. f(x) has a local minimum at x = -3, and

2. a local maximum at x = 3

Therefore f'(x) has to cross the x-axis at x = -3 and x=+3.

Furthermore, f'(x) must be increasing at x=-3 and decreasing at x=+3.

f'(x) = -x^2+9

will satisfy the above conditions.

Finally f(x) must be decreasing at x= -5, which implies that f'(-5) must be negative.

Check: f'(-5) = -(-5)^2+9 = -25+9 = -16 < 0  so ok.

f(x) can then be obtained by integrating f'(x) :

f(x) = integral of -x^2+9 = -x^3/3 + 9x = 9x - x^3/3

A graph of f(x) is attached, and is found to satisfy all three conditions.

step-by-step explanation:

descending order means to order the polynomials from the term with the largest exponent to the term with the smallest exponent.  

therefore, the correct answer is d.