Let f be the function given by f(x)=(1/7)x^7+12x^6−x^5−(15/4)x^4+(4/3)x^3+6x^2. Which of the following statements is true?
To determine which of the following statements is true for the function f(x)=(1/7)x^7+12x^6−x^5−(15/4)x^4+(4/3)x^3+6x^2, we need to analyze the function and its properties
To determine which of the following statements is true for the function f(x)=(1/7)x^7+12x^6−x^5−(15/4)x^4+(4/3)x^3+6x^2, we need to analyze the function and its properties.
Statement 1: The function f(x) is an even function.
An even function is symmetric with respect to the y-axis, meaning that f(x) = f(-x) for all x in the domain. To check if f(x) is an even function, we substitute -x into the function and compare it with f(x).
f(-x) = (1/7)(-x)^7 + 12(-x)^6 – (-x)^5 – (15/4)(-x)^4 + (4/3)(-x)^3 + 6(-x)^2
= (1/7)(-x)^7 + 12x^6 – x^5 – (15/4)x^4 – (4/3)x^3 + 6x^2
Comparing f(x) with f(-x), we can observe that f(x) is not equal to f(-x) for all x in the domain. Therefore, statement 1 is false.
Statement 2: The degree of the function f(x) is 7.
The degree of a polynomial is the highest power of x in that polynomial. Looking at the function f(x)=(1/7)x^7+12x^6−x^5−(15/4)x^4+(4/3)x^3+6x^2, we can see that the highest power of x is 7. Therefore, statement 2 is true.
Statement 3: The leading coefficient of the function f(x) is 1/7.
The leading coefficient is the coefficient of the term with the highest power of x. In the function f(x)=(1/7)x^7+12x^6−x^5−(15/4)x^4+(4/3)x^3+6x^2, the coefficient of the term with the highest power of x is 1/7. Therefore, statement 3 is true.
In conclusion, statement 2 and statement 3 are true for the function f(x)=(1/7)x^7+12x^6−x^5−(15/4)x^4+(4/3)x^3+6x^2.
More Answers:
How to Determine the Concavity of a Function and Interpret the ResultsA Guide to Curve Sketching: Analyzing the Behavior and Characteristics of Functions
Calculating the Difference between Approximation and Actual Value of f′(0.5) for the Function f(x) = 2x^3