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Is the function f(x) = x^3 - 3x + 2 differentiable at x = 1?

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Question: Is the function f(x) = x^3 - 3x + 2 differentiable at x = 1?

Options:

  1. Yes
  2. No
  3. Only left differentiable
  4. Only right differentiable

Correct Answer: Yes

Solution: 

The function is a polynomial and hence differentiable everywhere, including at x = 1.

Is the function f(x) = x^3 - 3x + 2 differentiable at x = 1?

Practice Questions

Q1
Is the function f(x) = x^3 - 3x + 2 differentiable at x = 1?
  1. Yes
  2. No
  3. Only left differentiable
  4. Only right differentiable

Questions & Step-by-Step Solutions

Is the function f(x) = x^3 - 3x + 2 differentiable at x = 1?
  • Step 1: Identify the function given, which is f(x) = x^3 - 3x + 2.
  • Step 2: Recognize that this function is a polynomial because it is made up of terms with x raised to whole number powers.
  • Step 3: Understand that polynomials are smooth and continuous functions.
  • Step 4: Know that all polynomial functions are differentiable everywhere on their domain.
  • Step 5: Since f(x) is a polynomial, conclude that it is differentiable at x = 1.
  • Differentiability of Polynomials – Polynomials are continuous and differentiable everywhere on their domain, which is all real numbers.
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