Graphing functions Use the guidelines of this section to make a complete graph of f.

f(x) = x³ - 6x² - 135x

검증된 단계별 안내

Step 1: Identify the domain of the function f(x) = x³ - 6x² - 135x. Since this is a polynomial function, the domain is all real numbers.

Step 2: Find the critical points by taking the derivative of f(x) to find f'(x). The derivative is f'(x) = 3x² - 12x - 135. Set f'(x) = 0 to find the critical points.

Step 3: Solve the equation 3x² - 12x - 135 = 0 to find the values of x where the function has critical points. Use the quadratic formula or factorization to find these values.

Step 4: Determine the behavior of the function at the critical points by using the second derivative test. Find f''(x) = 6x - 12 and evaluate it at the critical points to determine concavity.

Step 5: Analyze the end behavior of the function by considering the leading term x³. As x approaches positive or negative infinity, the function will behave like x³, which means it will go to positive infinity as x goes to positive infinity and negative infinity as x goes to negative infinity.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

10m

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주요 개념

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Polynomial Functions

A polynomial function is a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients. In this case, f(x) = x³ - 6x² - 135x is a cubic polynomial, which means its highest degree is three. Understanding the general shape and behavior of polynomial functions is crucial for graphing them effectively.

Critical Points and Extrema

Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection. To find these points for f(x), we need to compute its derivative, set it to zero, and solve for x. Analyzing these points helps in determining the overall shape and turning points of the graph.

End Behavior

End behavior describes how a function behaves as the input values approach positive or negative infinity. For polynomial functions, the leading term dictates this behavior. In the case of f(x), since the leading term is x³, the graph will rise to positive infinity as x approaches positive infinity and fall to negative infinity as x approaches negative infinity, shaping the overall graph.

Graphing functions Use the guidelines of this section to make a complete graph of f.f(x) = x³ - 6x² - 135x