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 determine the true statement about the function f, we can analyze its properties.
1. The degree of the function f is 7. This can be seen by the highest power of x in the expression, which is x^7.
2. The coefficient of the x^7 term is 1/7. This tells us that the function has a positive leading coefficient.
3. The coefficient of the x^6 term is 12. This is a positive coefficient, which means that the function increases rapidly as x increases.
4. The coefficient of the x^5 term is -1, which means that the function changes direction at some point. Specifically, it has a local maximum or minimum at x=0.
5. The coefficient of the x^4 term is negative (-15/4), which means that the function decreases as x increases.
6. The coefficient of the x^3 term is positive (4/3), which means that the function increases as x increases.
7. The coefficient of the x^2 term is positive, which means that the function increases rapidly as x increases.
Based on these properties, we can conclude that the true statement is:
The function f has a local maximum or minimum at x=0, and increases rapidly as x increases for large values of x.
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To determine the true statement about the function f, we can analyze its properties.
1. The degree of the function f is 7. This can be seen by the highest power of x in the expression, which is x^7.
2. The coefficient of the x^7 term is 1/7. This tells us that the function has a positive leading coefficient.
3. The coefficient of the x^6 term is 12. This is a positive coefficient, which means that the function increases rapidly as x increases.
4. The coefficient of the x^5 term is -1, which means that the function changes direction at some point. Specifically, it has a local maximum or minimum at x=0.
5. The coefficient of the x^4 term is negative (-15/4), which means that the function decreases as x increases.
6. The coefficient of the x^3 term is positive (4/3), which means that the function increases as x increases.
7. The coefficient of the x^2 term is positive, which means that the function increases rapidly as x increases.
Based on these properties, we can conclude that the true statement is:
The function f has a local maximum or minimum at x=0, and increases rapidly as x increases for large values of x.
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