Consider a polynomial function 𝑓 ( 𝑥 ) = − 1 300 ( 𝑥 − 2 ) 2 ( 𝑥 − 3 ) ( 𝑥 + 1 ) 2 ( 𝑥 + 4 ) ( 𝑥 − 5 ) f(x)= 300 −1 (x−2) 2 (x−3)(x+1) 2 (x+4)(x−5). Choose the correct set of options. The function 𝑓 ( 𝑥 ) f(x) has exactly 7 turning points. The function 𝑓 ( 𝑥 ) f(x) has exactly 6 points where the slope is 0. The function 𝑓 ( 𝑥 ) f(x) is neither even nor odd function. In the interval 𝑥 ∈ ( 3 , 5 ) x∈(3,5), 𝑓 ( 𝑥 ) f(x) is increasing first and then decreasing. The function 𝑓 ( 𝑥 ) f(x) is negative when 𝑥 ∈ ( − 1 , 2 ) x∈(−1,2).
Question
Consider a polynomial function 𝑓 ( 𝑥 )
− 1 300 ( 𝑥 − 2 ) 2 ( 𝑥 − 3 ) ( 𝑥 + 1 ) 2 ( 𝑥 + 4 ) ( 𝑥 − 5 ) f(x)= 300 −1 (x−2) 2 (x−3)(x+1) 2 (x+4)(x−5). Choose the correct set of options.
The function 𝑓 ( 𝑥 ) f(x) has exactly 7 turning points.
The function 𝑓 ( 𝑥 ) f(x) has exactly 6 points where the slope is 0.
The function 𝑓 ( 𝑥 ) f(x) is neither even nor odd function.
In the interval 𝑥 ∈ ( 3 , 5 ) x∈(3,5), 𝑓 ( 𝑥 ) f(x) is increasing first and then decreasing.
The function 𝑓 ( 𝑥 ) f(x) is negative when 𝑥 ∈ ( − 1 , 2 ) x∈(−1,2).
Solution
The function f(x) is a polynomial of degree 7 (since the highest power of x is 7).
-
The function f(x) has exactly 7 turning points: This statement is incorrect. A polynomial of degree n has at most n-1 turning points. So, a 7th degree polynomial has at most 6 turning points.
-
The function f(x) has exactly 6 points where the slope is 0: This statement is correct. The derivative of the function gives the slope, and setting the derivative equal to zero gives the critical points. A polynomial of degree n has at most n critical points, so a 7th degree polynomial has at most 7 critical points. However, since some of the factors in the polynomial are squared, they will not contribute to the count of unique critical points, so there will be fewer than 7.
-
The function f(x) is neither even nor odd function: This statement is correct. An even function is symmetric about the y-axis, and an odd function is symmetric about the origin. This function is not symmetric in either of these ways.
-
In the interval x∈(3,5), f(x) is increasing first and then decreasing: Without doing specific calculations, it's hard to definitively say whether this is true or false. However, given the nature of polynomial functions, it is plausible.
-
The function f(x) is negative when x∈(−1,2): Again, without specific calculations, it's hard to definitively say. However, given the nature of polynomial functions, it is plausible.
Similar Questions
Determine the end behavior, maximum number of turning point(s), and maximum number ofintercepts of the following polynomial functions:(a) .(b) .(c) .(d)(e)(f)XAVIER SCHOOLSenior High School Standard ProgramGRADE 11 GENERAL MATHEMATICSSecond Semester, S.Y. 2023-2024x−P(x) = − x 3 + 20x 2 + x + 10P(x) = 15x 3 − 18x 2 + 4x − 8P(x) = 13x 4 − 2x 3 + 5x 2 + x − 6P(x) = (x + 1)(3 − x)(3x − 1)(4x + 1)P(x) = (2x − 1) 2 (1 − x)(5 − 3x)2P(x) = − (x − 1)(x + 3)2 (2 + 3x)3
Consider the part of a graph of a polynomial, p(x), which is shown. Which of the following could be functions that match the graph of p(x)? Choose all that apply:(1 Point)p(x) = (x - 5)(x + 3)(x + 6)p(x) = (x + 5)(x - 3)(x - 6)p(x) = (x - 5)(x + 180)(x + 3)(x + 6)p(x) = (x + 5)(x - 180)(x - 3)(x - 6)None of the above could match the graph of p(x)
1. A polynomial function can have more than one y-intercepts. (True/False)2. An increasing polynomial function is always one-to-one. (True/False)3. A polynomial function of degree s can have s + 1 zeros. (True/False)4. There is a turning point between any two zeros of a polynomial function.(True/False)5. If a critical point (turning point) of a polynomial function p(x) exists,then p(x) is not one-to-one. (True/False)
The graph of a polynomial p(x) cuts the X-axis at 3 points and touches it at 2 other points. It also touches the Y-axis at 1 point. The number of zeroes of p(x) is
Given x=2 and x=-1 anre zeros with multiplicity 1 and two respectively of a polynomimal function of degree 3. If f(0) = 6, then f(x)=A.(x+1)2(x-2)B.-3(x+1)2(x+2)C.3(x+1)2(x-2)D.-3(x+1)2(x-2)
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.