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, we need to analyze the given function f(x) and its properties
To determine which of the following statements is true, we need to analyze the given function f(x) and its properties.
The function f(x) is a polynomial function of degree 7. This means that the function is defined for all real numbers. We can analyze the behavior of the function by looking at the leading coefficient and the signs of the coefficients.
Let’s break down the function f(x) and examine its different terms:
f(x) = (1/7)x^7 + 12x^6 − x^5 − (15/4)x^4 + (4/3)x^3 + 6x^2
– The leading term is (1/7)x^7, which has a positive coefficient (1/7). This indicates that the function increases without bound as x approaches positive infinity and decreases without bound as x approaches negative infinity.
– The coefficient of the term 12x^6 is positive, which means that the function has a positive steepness or slope for large values of x.
– The coefficient of the term -x^5 is negative, which means that the function has a negative steepness or slope for values of x between zero and negative infinity.
– The coefficient of the term -(15/4)x^4 is negative, which means that the function has a negative steepness or slope for large negative values of x.
– The coefficient of the term (4/3)x^3 is positive, which means that the function has a positive steepness or slope for large values of x.
– The coefficient of the term 6x^2 is positive, which means that the function has a positive steepness or slope for all values of x.
Based on the analysis of the function f(x), we can conclude that the statement “The function f(x) is increasing for all real numbers.” is false. Since the function has terms with both positive and negative coefficients, its behavior varies for different intervals of x.
Therefore, the correct answer is that none of the statements is true.
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