Math Problem Statement

What is f’(-4)

What is f'(-4)?

Solution

To determine f(4)f'(-4), we need to find the slope of the graph at x=4x = -4, as f(x)f'(x) represents the derivative of f(x)f(x), which is the slope of the tangent line to the curve at a given point.

From the graph, between x=5x = -5 and x=3x = -3, the function appears to be a straight line. Specifically:

  • At x=5x = -5, f(x)=3f(x) = 3
  • At x=3x = -3, f(x)=3f(x) = 3

This means the line segment is horizontal between these points, so the slope of the line between x=5x = -5 and x=3x = -3 is 0. Therefore, the derivative at x=4x = -4, which lies on this horizontal segment, is:

f(4)=0f'(-4) = 0

Would you like more details or have any further questions?

Here are some follow-up questions related to this graph:

  1. What is f(1)f'(1)?
  2. What is the value of f(2)f(-2)?
  3. Where is the function f(x)f(x) increasing?
  4. Identify any discontinuities in the graph of f(x)f(x).
  5. What is the maximum value of f(x)f(x)?

Tip: The slope of a line segment is calculated by dividing the difference in yy-values by the difference in xx-values for two points on the line.

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Math Problem Analysis

Mathematical Concepts

Calculus
Derivative
Slope of a Line

Formulas

Slope Formula: m = (y2 - y1) / (x2 - x1)

Theorems

Basic Derivative Theorem

Suitable Grade Level

Grades 10-12