Answer:
a) the intervals where f is increasing is ![(-\infty,-1) \cup(\dfrac{1}{2},\infty)](https://tex.z-dn.net/?f=%28-%5Cinfty%2C-1%29%20%5Ccup%28%5Cdfrac%7B1%7D%7B2%7D%2C%5Cinfty%29)
the intervals where f is decreasing is ![(-1,\dfrac{1}{2})](https://tex.z-dn.net/?f=%28-1%2C%5Cdfrac%7B1%7D%7B2%7D%29)
b)
: local maximum
: local minimum
c) The inflection point : ![x = 0.25](https://tex.z-dn.net/?f=x%20%3D%200.25)
the range of concavity is: ![(-\infty,0.25)](https://tex.z-dn.net/?f=%28-%5Cinfty%2C0.25%29)
Step-by-step explanation:
This is a positive cubic function, (positive only means that the sign of the highest power term is +ve, nothing too fancy, its just to visualize the shape of the curve)
<em>the positive sign tells you that this curve is coming from negative y to positive y when looking from left to right. </em>
![f(x) = 4x^3 +3x^2 -6x +4](https://tex.z-dn.net/?f=f%28x%29%20%3D%204x%5E3%20%2B3x%5E2%20-6x%20%2B4)
we can find the intervals where it is increasing and decreasing by knowing where this function has its stationary points (or turning points), in other words where ![f'(x) = 0](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%200)
![f(x) = 4x^3 +3x^2 -6x +4](https://tex.z-dn.net/?f=f%28x%29%20%3D%204x%5E3%20%2B3x%5E2%20-6x%20%2B4)
![f'(x) = 12x^2 +6x -6](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%2012x%5E2%20%2B6x%20-6)
this is the function's first derivative. To find the stationary values, set ![f'(x) = 0](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%200)
![0 = 12x^2 +6x -6](https://tex.z-dn.net/?f=0%20%3D%2012x%5E2%20%2B6x%20-6)
now solve for x
![0 = (2x-1)(x+1)](https://tex.z-dn.net/?f=0%20%3D%20%282x-1%29%28x%2B1%29)
![x = 1/2,\,\, x=-1](https://tex.z-dn.net/?f=x%20%3D%201%2F2%2C%5C%2C%5C%2C%20x%3D-1)
we can find the local minimum and maximum values of f by plugging in these value in the original function f(x):
![f(x) = 4x^3 +3x^2 -6x +4](https://tex.z-dn.net/?f=f%28x%29%20%3D%204x%5E3%20%2B3x%5E2%20-6x%20%2B4)
local maximum
local minimum
We also have enough information to show the intervals at which f(x) is increasing or decreasing
Since this is a positive cubic curve, the plot is coming up from negative infinity of the y-axis all the way upto x= -1, then turns back down until it reaches x=1/2, then finally turns up again to positive infinity of the y-axis.
so,
the intervals where f is increasing is ![(-\infty,-1) \cup(\dfrac{1}{2},\infty)](https://tex.z-dn.net/?f=%28-%5Cinfty%2C-1%29%20%5Ccup%28%5Cdfrac%7B1%7D%7B2%7D%2C%5Cinfty%29)
the intervals where f is decreasing is ![(-1,\dfrac{1}{2})](https://tex.z-dn.net/?f=%28-1%2C%5Cdfrac%7B1%7D%7B2%7D%29)
Concavity and Inflection Points
Now, we can go further in by differentiating our
![f'(x) = 12x^2 +6x -6](https://tex.z-dn.net/?f=f%27%28x%29%20%3D%2012x%5E2%20%2B6x%20-6)
![f''(x) = 24x +6](https://tex.z-dn.net/?f=f%27%27%28x%29%20%3D%2024x%20%2B6)
we can put the values we obtained of x from
to find the curvature(or shape) of the curve at those points
![f''(-1) = -18](https://tex.z-dn.net/?f=f%27%27%28-1%29%20%3D%20-18%20)
this is a negative value, it shows that at this value of x, the curve looks like this:
(this is known a concave shape)
![f''(\dfrac{1}{2}) = 18](https://tex.z-dn.net/?f=f%27%27%28%5Cdfrac%7B1%7D%7B2%7D%29%20%3D%2018%20)
this this is a positive value, it shows that at this value of x, the curve looks like this:
(this is knows as the convex shape)
with this much information we have some idea about the concavity.<em> (i.e, for what range of x does the curve maintain
shape and for what range of x the curve maintains
shape?)</em>
we know that for the intervals
the curve is increasing, <em>but </em>the shape remains like
even after this range.
So what we need is a point where the two shapes begin to change:
and that is the inflection point:
to put in terms of math: the inflection point is where: ![f''(x) = 0](https://tex.z-dn.net/?f=f%27%27%28x%29%20%3D%200)
![f''(x) = 24x +6](https://tex.z-dn.net/?f=f%27%27%28x%29%20%3D%2024x%20%2B6)
![0 = 24x +6](https://tex.z-dn.net/?f=0%20%3D%2024x%20%2B6)
![x = -0.25](https://tex.z-dn.net/?f=x%20%3D%20-0.25%20)
this is the point where concave turns to convex.
the inflection point is: ![x = 0.25](https://tex.z-dn.net/?f=x%20%3D%200.25)
the range of concavity is: ![(-\infty,0.25)](https://tex.z-dn.net/?f=%28-%5Cinfty%2C0.25%29)
for fun we can also find the range of convexity
the range of convexity is: ![(0.25, \infty)](https://tex.z-dn.net/?f=%280.25%2C%20%5Cinfty%29)
hopefully, this was helpful and a fun read.