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?
f′(0.4)
To answer this question, we need to consider the properties of the given function f(x).
First, let’s determine the degree of f(x). The degree of a polynomial function is the highest power of x that appears in the function. In this case, the highest power of x is 7, so the degree of f(x) is 7.
Next, let’s consider the leading coefficient of f(x). The leading coefficient is the coefficient of the term with the highest power of x. In this case, the leading coefficient is 1/7.
Using this information, we can make the following conclusions:
1. The function f(x) is a polynomial function of degree 7.
2. The leading coefficient of f(x) is positive (1/7), so the end behavior of the function is as follows:
– As x approaches negative infinity, f(x) approaches negative infinity since a positive number divided by a very large negative number is a very small negative number.
– As x approaches positive infinity, f(x) approaches positive infinity since a positive number divided by a very large positive number is a very small positive number.
3. Since the degree of f(x) is odd (7), the graph of f(x) will cross the x-axis at least once. This means that f(x) has at least one real root.
Therefore, the statement that is true is: The function f(x) has at least one real root.
More Answers:
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Find the Equation of the Line Tangent to the Graph of f at x=-1 – A Step-by-Step Guide.
Calculating the Average Rate of Change of a Function f(x) Over a Specified Interval [0,π]
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To answer this question, we need to consider the properties of the given function f(x).
First, let’s determine the degree of f(x). The degree of a polynomial function is the highest power of x that appears in the function. In this case, the highest power of x is 7, so the degree of f(x) is 7.
Next, let’s consider the leading coefficient of f(x). The leading coefficient is the coefficient of the term with the highest power of x. In this case, the leading coefficient is 1/7.
Using this information, we can make the following conclusions:
1. The function f(x) is a polynomial function of degree 7.
2. The leading coefficient of f(x) is positive (1/7), so the end behavior of the function is as follows:
– As x approaches negative infinity, f(x) approaches negative infinity since a positive number divided by a very large negative number is a very small negative number.
– As x approaches positive infinity, f(x) approaches positive infinity since a positive number divided by a very large positive number is a very small positive number.
3. Since the degree of f(x) is odd (7), the graph of f(x) will cross the x-axis at least once. This means that f(x) has at least one real root.
Therefore, the statement that is true is: The function f(x) has at least one real root.
More Answers:
Why f(2) Exists Only When f is Continuously Defined at x=2: the Relationship Between Continuity and Function ExistenceFind the Equation of the Line Tangent to the Graph of f at x=-1 – A Step-by-Step Guide.
Calculating the Average Rate of Change of a Function f(x) Over a Specified Interval [0,π]