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What is the definition of a polynomial function?

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A polynomial function is a function that can be expressed in the form f(x) = a_n*x^n + a_(n-1)*x^(n-1) + ... + a_1*x + a_0, where a_n, a_(n-1), ..., a_0 are constants and n is a non-negative integer.

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What does it mean to find the maximum or minimum of a function?

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Finding the maximum or minimum of a function involves determining the highest or lowest point on the graph of the function, which can be found using calculus techniques such as taking the derivative and setting it to zero.

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How do you find the vertex of a cubic function?

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The vertex of a cubic function can be found by using calculus to find the critical points, or by analyzing the function's behavior and identifying the point where the function changes direction.

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What are the intervals where a polynomial function is negative?

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To find the intervals where a polynomial function is negative, you need to determine the zeros of the function and test the sign of the function in the intervals created by these zeros.

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What is a zero of a function?

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A zero of a function is a value of x for which f(x) = 0. It represents the x-intercepts of the graph of the function.

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How do you determine the zeros of a polynomial function?

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The zeros of a polynomial function can be determined by factoring the polynomial, using the Rational Root Theorem, or applying numerical methods such as synthetic division or the quadratic formula.

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What is the significance of the leading coefficient in a polynomial function?

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The leading coefficient determines the end behavior of the polynomial function. If it is positive, the function will rise to the right; if negative, it will fall to the right.

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What is the difference between a local maximum and a global maximum?

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A local maximum is the highest point in a specific interval, while a global maximum is the highest point over the entire domain of the function.

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What is the role of the first derivative in finding extrema of a function?

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The first derivative of a function indicates the slope of the function. Setting the first derivative to zero helps identify critical points where the function may have local maxima or minima.

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What is the purpose of the second derivative test?

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The second derivative test is used to determine the concavity of a function at critical points. If the second derivative is positive, the point is a local minimum; if negative, it is a local maximum.