Math Problem Statement

Problem

Consider the polynomial function:

f(x)=x37x2+14x8f(x) = x^3 - 7x^2 + 14x - 8

  1. Find the first derivative of ( f(x) ).

  2. Factor the derivative.

  3. Use the factored derivative to determine the points where the slope of the function is zero (i.e., the critical points).

Solution

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Math Problem Analysis

Mathematical Concepts

Differentiation
Quadratic Equations
Factoring Polynomials

Formulas

Power rule for derivatives: d/dx(x^n) = nx^(n-1)
Quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a

Theorems

Quadratic formula

Suitable Grade Level

Grade 10-12