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?
10 at most
seven at most
two at most
eight at most
The turning points of a graph correspond to the points where the graph changes from increasing to decreasing or from decreasing to increasing. These points occur where the derivative of the function is equal to zero.
To find the number of turning points of the function m(x), we need to find the derivative of m(x) and count the number of zeros of the derivative.
Taking the derivative of m(x) using the product rule and chain rule, we get:
m'(x) = (2x+5)(3(x-3)^2)(8x+1)^5 + (2x+5)(x-3)^3(5(8x+1)^4)(8) + (2)(x-3)^3(8x+1)^5
Simplifying this, we get:
m'(x) = (2x+5)(3(x-3)^2)(8x+1)^5 + 8(2x+5)(x-3)^3(8x+1)^4 + 2(x-3)^3(8x+1)^5
Now we need to find the zeros of m'(x). Since m'(x) is a polynomial, we can use any method that finds the roots of polynomials, such as factoring, graphing, or using the rational root theorem. However, for simplicity, we can plug in some values of x to see if m'(x) equals zero at any of these values:
m'(-2) = (2(-2)+5)(3((-2)-3)^2)(8(-2)+1)^5 + 8(2(-2)+5)((-2)-3)^3(8(-2)+1)^4 + 2((-2)-3)^3(8(-2)+1)^5
= (1)(3)(-15)^5 + 8(1)(-5)(-15)^4 + (-5)^3(-15)^5
= -1125 + 14400 - 16875
= -2250
m'(0) = (2(0)+5)(3((0)-3)^2)(8(0)+1)^5 + 8(2(0)+5)((0)-3)^3(8(0)+1)^4 + 2((0)-3)^3(8(0)+1)^5
= (5)(3)(-9)^5 + 8(5)(-3)^4 + (-3)^3
= -2430 + 1440 - 27
= -1017
m'(1) = (1)(3)(8)^5 + 8(1)(-2)^4 + (-2)^3
= 12288 + 128 - 8
= 12408
From these calculations, we can see that m'(-2) and m'(0) are negative while m'(1) is positive. This suggests that m'(x) has at least 2 zeros, indicating the presence of at least 2 turning points in the graph of m(x).
Therefore, the correct answer is: two at most.
To determine the number of turning points for the graph of m(x), we need to examine the degree and multiplicity of each factor in the function.
For the factor (2x+5), it has a degree of 1 and a multiplicity of 1. This means it will contribute at most 1 turning point.
For the factor (x-3)^3, it has a degree of 3 and a multiplicity of 3. This means it will contribute at most 3 turning points.
For the factor (8x+1)^5, it has a degree of 5 and a multiplicity of 5. This means it will contribute at most 5 turning points.
To find the total number of turning points, we add up the maximum number of turning points contributed by each factor:
1 (2x+5) + 3 (x-3)^3 + 5 (8x+1)^5 = 1 + 3 + 5 = 9
Since the product of the factors can potentially create 9 turning points at most, the correct answer is eight at most.