Brandy is asked to sketch a graph of m(x)=(2x+5)(x−3)3(8x+1)5
. How many turning points should her sketch include?(1 point)
Responses
eight at most
eight at most
seven at most
seven at most
10 at most
10 at most
two at most
The correct response is: two at most.
To determine the number of turning points on the graph of the function m(x)=(2x+5)(x−3)³(8x+1)⁵, we need to consider the powers of each factor.
The factor (x−3)³ has odd power, which means it will have one turning point.
The factor (8x+1)⁵ has odd power, which means it will have one turning point.
The remaining factor (2x+5) does not affect the number of turning points because it has an even power.
Therefore, the total number of turning points will be two at most.
To determine the number of turning points on the graph of the given function m(x), we need to analyze the behavior of the function and identify where it changes direction.
To do this, we can start by finding the derivative of the function m(x) and then determining the number of critical points. Critical points are the values of x where the derivative is equal to zero or is undefined.
1. Find the derivative of m(x):
Take the derivative of each term separately using the power rule and product rule:
m'(x) = (2x+5)'(x−3)³(8x+1)⁵ + (2x+5)(x−3)³(8x+1)⁵'
2. Simplify and solve for m'(x) = 0:
Simplify the derivative equation obtained in step 1 and solve for m'(x) = 0 to find the critical points. This will give us the x-coordinates of the turning points.
3. Determine the number of turning points:
Count the number of distinct critical points obtained in step 2. Each distinct critical point represents a potential turning point on the graph of m(x).
In this case, since the function m(x) is given as a product of different terms, which are raised to different powers, finding the derivative and solving for critical points can become complex. You can use a graphing calculator or software to plot the graph of m(x) and visually identify the number of turning points.
It is important to note that the number of turning points mentioned in the options provided (eight at most, seven at most, ten at most, two at most) are all possibilities. However, without further analysis, it is difficult to determine the precise number of turning points without any additional context or information about the function.