SOLUTION: Sketch the graph of a single function that has all of the properties listed. a. Continuous and differentiable for all real numbers b. f'(x)<0 on (-∞,-4) and(0,5) c. f'(x)>0 on

Question 1195393: Sketch the graph of a single function that has all of the properties listed.
a. Continuous and differentiable for all real numbers
b. f'(x)<0 on (-∞,-4) and(0,5)
c. f'(x)>0 on (-4,0) and (5,∞)
d. f''(x)>0 on (-∞,-1) and (2,∞)
e. f''(x)<0 on (-1,2)
f. f'(-4)=f'(5)=0
g.f''(x)=0 at (-1,11) and (2,10)
any help will be so appreciated this is the only question i have left on my homework and im stumped):
thank you so much in advance!
Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

--------------------------------------------------------------------

I need to rewrite my response -- in the question box the derivatives and second derivatives did not show up well, so originally I was working with all the wrong conditions....

--------------------------------------------------------------------

Note the problem does not ask you to find an equation -- it only asks you to sketch a graph.

But even with that I am still stumped, like you, because I don't think the given conditions can be met.

(1) Conditions b and c tell us that the graph has local minima at x=-4 and x=5, and a local maximum at x=0. Note that means that condition f is superfluous.

(2) Conditions d and e tell us the concavity changes from upwards to downwards at x=-1 and from downwards to upwards at x=2, so there are points of inflection at x=-1 and x=2.

So conditions b, c, d, e, and f give us 3 local extrema and 2 points of inflection, suggesting a polynomial of degree 4.

However, all of the given conditions can't be met with a polynomial of degree 4.

Given local extrema ONLY at x=-4, x=0, and x=5, the equation of the derivative is

f'(x) =

Then the second derivative would be

f''(x) =

But that second derivative does not have zeros at x=-1 and x=2.

So it appears the best we can do with this problem is to ignore some of the conditions, and -- without trying to find an equation -- sketch a graph that is...
concave up from -infinity to -1, with a local minimum at -4;
concave down from -1 to 2, with a local maximum at 0; and
concave up from to infinity, with a local minimum at x=5.

----------------------------------------------------------------

Perhaps you will get other responses to your post from tutors who see the problem differently than I do....

RELATED QUESTIONS
Sketch the graph of a single function that has all of the properties listed. a.... (answered by ElectricPavlov)

Piecewise function. f(x)={x+4, x ≥ 2 and ax^2, x < 2 A. Find the value for "a" (answered by solver91311)

Determine all the pairs of real numbers a and b so that the following function, f(x) is... (answered by Edwin McCravy)

If f(x) is continuous and differentiable real valued function defined on open interval... (answered by ikleyn)

Sketch the graph of a function f(x) that satisfies the stated condition. Mark any... (answered by jim_thompson5910)

1. (7 points) On the axes below, sketch the graph of one possible function y = f(x) be a... (answered by greenestamps)

Give an example of a function which is differentiable at all points except 1,3 and... (answered by ikleyn)

Sketch a graph of a function f that satisfies the following conditions: f does NOT... (answered by robertb)

Give an example of a function which is differentiable at all points except... (answered by ikleyn)