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If a function f(x) is defined as f(x) = x^3 - 3x + 2, what can be inferred about

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Question: If a function f(x) is defined as f(x) = x^3 - 3x + 2, what can be inferred about its behavior at critical points?

Options:

  1. It has no critical points.
  2. It has one local maximum and one local minimum.
  3. It is always increasing.
  4. It is always decreasing.

Correct Answer: It has one local maximum and one local minimum.

Solution: 

To find critical points, we take the derivative and set it to zero. The function has one local maximum and one local minimum based on the nature of cubic functions.

If a function f(x) is defined as f(x) = x^3 - 3x + 2, what can be inferred about

Practice Questions

Q1
If a function f(x) is defined as f(x) = x^3 - 3x + 2, what can be inferred about its behavior at critical points?
  1. It has no critical points.
  2. It has one local maximum and one local minimum.
  3. It is always increasing.
  4. It is always decreasing.

Questions & Step-by-Step Solutions

If a function f(x) is defined as f(x) = x^3 - 3x + 2, what can be inferred about its behavior at critical points?
  • Step 1: Identify the function given, which is f(x) = x^3 - 3x + 2.
  • Step 2: Find the derivative of the function, which is f'(x). This tells us the slope of the function.
  • Step 3: Set the derivative equal to zero (f'(x) = 0) to find critical points. These are the x-values where the function's slope is zero.
  • Step 4: Solve the equation from Step 3 to find the x-values of the critical points.
  • Step 5: Determine whether each critical point is a local maximum, local minimum, or neither by using the second derivative test or analyzing the first derivative around those points.
  • Step 6: Conclude that the function has one local maximum and one local minimum based on the behavior of cubic functions.
  • Critical Points – Critical points occur where the derivative of a function is zero or undefined, indicating potential local maxima, minima, or points of inflection.
  • Cubic Functions – Cubic functions can have up to two turning points, which can be identified through their first and second derivatives.
  • Local Extrema – Local maxima and minima are determined by analyzing the sign of the first derivative around critical points.
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