How do you know this treaky question?

Question

Given f(x)=x^4-7x^3-6x^2+8x+9.
Determine the coordinates of the lowest point at which the graph changes direction from decreasing to increasing.

Answer 1

Ok, simple differentiation problem, right?

First derivative = 0 and solve for x will tell you the x-ordinates for all peaks and valleys. There should be three.

Second derivative at each of those points will tell you whether the graph is concave downward (-) or concave upward (+).
Concave downward means a change from rising to falling
Concave upward means a change from falling to rising.
Second derivative = 0 means that is a reflex point, the graph changes from rising to flat to riasing or from falling to flat to falling.

That is a general rule for any degree of polynomial.

Answer 2

Differentiate and equate to zero.
f(x) = x^4 - 7x^3 - 6x^2 + 8x +9
f'(x) = 4x^3 - 21x^2 - 12x + 8 = 0
By trial and error the answer for 'x' lies between '0' and '1'. Because when 'x' is '0' f'(x) = +8 and when 'x' is '1' f'(x) = -21.
So between 0 and 1 the differentiated function changes from +ve to -ve, through zero(0).
The value for 'x' at zero, will be the minimum point.
To find the 'y' coordinate take the value for 'x' and substitute it bback in to the f(x) eq'n.
This will then given you the 'y' coordinate.at the minimum point.

Answer 3

Egads! You got to solve g(x) = 4x^3 - 21x^2 - 12x + 8 = 0
Then plug the 3 values of x into f(x). Or actually, the middle value of X will be for f(x) increasing to decreasing, so you can skip that one.

Answer 4

Hint; Solve for slope = 0