Slope equations

[asu11]

f(x)=x^3-x^2+2x-17
g(x)=-4.5x^2-2x+3

a) At what value(s) of x are the slope equations for f(x) and g(x) the same value?

my work:
x^3-x^2+2x-17=-4.5x^2-2x+3
x^3+3/5x^2+4x=20
thats as far as I have gotten

b) The concavities for f(x) and g(x) are the same value at exactly one value of x; is that common cavity up, down, or an inflection point?

 

[wjm11]

f(x)=x^3-x^2+2x-17
g(x)=-4.5x^2-2x+3

a) At what value(s) of x are the slope equations for f(x) and g(x) the same value?

my work:
x^3-x^2+2x-17=-4.5x^2-2x+3
x^3+3/5x^2+4x=20
thats as far as I have gotten

b) The concavities for f(x) and g(x) are the same value at exactly one value of x; is that common cavity up, down, or an inflection point?

For part a) you need to set the derivatives equal to each other, not the original functions.

For part b) you need to examine the second derivatives. When second derivatives are positive, the functions are concave upward -- and vice versa for negative.

 

[swaroop]

Slope of a function= first derivative of function
Slope of f(x)=f'(x)=3x^2-2x+2
Slope of g(x)=g'(x)=-9x-2
f'(x)=g'(x) solve for this x to get the answer

b) Take second derivative and equalte them
f''(x)=6x-2
g''(x)=-9

 

[BigGlenntheHeavy]

a) See Graph:

[attachment=3:3b0vg9lw]abc.jpg[/attachment:3b0vg9lw]

b) Concave down (-infinity,1/3], see graph, inflection point for f(x):

\(\displaystyle g(x) \ is \ concave \ down \ for \ (-\infty,\infty) \ and \ f(x) \ is \ concave \ down \ for \ (-\infty,1/3]\)

\(\displaystyle What \ is \ with \ the \ unique \ x?\)

[attachment=2:3b0vg9lw]def.jpg[/attachment:3b0vg9lw]

\(\displaystyle b) \ Unique \ X. \ x \ = \ -7/6\)

\(\displaystyle See \ graph, \ note \ green \ line \ represents \ y \ = \ -9, \ both \ functions \ concave \ down.\)

[attachment=1:3b0vg9lw]ghi.jpg[/attachment:3b0vg9lw]

\(\displaystyle b) \ Graph \ of \ all \ three \ with \ their \ respective \ derivatives.\)

[attachment=0:3b0vg9lw]jkl.jpg[/attachment:3b0vg9lw]