Answer:
The following will help you plot the graph and answer the questions about its characteristics.
Explanation:
f(x) = x^3 - 10x^2 + 28x - 24
As the leading coefficient is 1 (positive) the graph will descend to the left and rise to the right.
Relative maximum / minimum:
Find the derivative:
f'(x) = 3x^2 - 20x + 28 = 0 ( for turning points).
(3x - 14 )(x - 2) = 0
x = 14/3 , 2.
The second derivative is 6x - 20.
When x = 14/3 second derivative is positive.
When x = 2 second derivative is negative
so x = 14/3 gives a relative minimum value for f(x)
and x = 2 gives a relative maximum.
If you plug these 2 values into f(x) you'll get the corresponding values of f(x) and you can plot theses points.
The y-intercept is where x = 0 so it's the point (0, -24) so plot this as well.
If there is one root which has multiplicity 2 then the graph will just touch the x-axis. This will be either one of the above : relative maximum or minimum.