Given a function f(x) = 4x^3 + 3x^2 − 6x + 1 (a) find the intervals on which f is increasing or decreasing. (b) find the local maximum and minimum values of f. (c) find the intervals of concavity and the inflection points.

a) the intervals where f is increasing is

the intervals where f is decreasing is

b) : local maximum

: local minimum

c) The inflection point :  

the range of concavity is:

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)

the positive sign tells you that this curve is coming from negative y to positive y when looking from left to right.

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

this is the function's first derivative. To find the stationary values, set

now solve for x

we can find the local minimum and maximum values of f by plugging in these value in the original function f(x):

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

the intervals where f is decreasing is

Concavity and Inflection Points

Now, we can go further in by differentiating our  

we can put the values we obtained of x from to find the curvature(or shape) of the curve at those points

this is a negative value, it shows that at this value of x, the curve looks like this: (this is known a concave shape)

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. (i.e, for what range of x does the curve maintain shape and for what range of x the curve maintains shape?)

we know that for the intervals the curve is increasing, but 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:

this is the point where concave turns to convex.

the inflection point is:  

the range of concavity is:

for fun we can also find the range of convexity

the range of convexity is:

hopefully, this was helpful and a fun read.

yeet

step-by-step explanation:

answer: the answer is (b) amie should have multiplied

step-by-step explanation: given a sphere and a cylinder with same radius and height.

let, 'r' be the radius of the cylinder and sphere and 'h' be the height of the cylinder. according to the question,

r = h.

now, the volume of the sphere will be

and the volume of the cylinder will be

therefore, we have

thus, ammie should have multiplied the volume of the sphere