Math Problem Statement

Let f(x) be given by: f(x) = {x + 1 if x < 1, x^2 + cx + d - c if x >= 1}. For which values of c and d is f(x) differentiable at x = 1?

Solution

To determine the values of cc and dd such that f(x)f(x) is differentiable at x=1x = 1, we need to ensure that two conditions hold:

  1. Continuity at x=1x = 1: The left-hand limit f(1)f(1^-) must equal the right-hand limit f(1+)f(1^+).
  2. Differentiability at x=1x = 1: The left-hand derivative f(1)f'(1^-) must equal the right-hand derivative f(1+)f'(1^+).

Step 1: Continuity at x=1x = 1

The function is defined as:

\begin{cases} x + 1 & \text{if } x < 1, \\ x^2 + cx + d - c & \text{if } x \geq 1. \end{cases}$$ For continuity at $$x = 1$$, we need to ensure that the function values from both pieces match at $$x = 1$$. - From the first piece when $$x = 1$$: $$f(1^-) = 1 + 1 = 2.$$ - From the second piece when $$x = 1$$: $$f(1^+) = 1^2 + c(1) + d - c = 1 + c + d - c = 1 + d.$$ For continuity, we set these equal: $$1 + d = 2.$$ This gives: $$d = 1.$$ ### Step 2: Differentiability at $$x = 1$$ Next, we calculate the derivatives of both pieces of the function and ensure they are equal at $$x = 1$$. - For $$x < 1$$, the derivative of $$f(x) = x + 1$$ is: $$f'(x) = 1.$$ Thus, $$f'(1^-) = 1$$. - For $$x \geq 1$$, the derivative of $$f(x) = x^2 + cx + d - c$$ is: $$f'(x) = 2x + c.$$ Thus, $$f'(1^+) = 2(1) + c = 2 + c$$. For differentiability, we set these derivatives equal: $$2 + c = 1.$$ This gives: $$c = -1.$$ ### Final Answer: The function $$f(x)$$ is differentiable at $$x = 1$$ when $$c = -1$$ and $$d = 1$$. Would you like further details or clarifications? Here are some related questions you might explore: 1. How does ensuring continuity affect differentiability at a point? 2. Can you generalize the process of checking differentiability for piecewise functions? 3. What happens if $$f(x)$$ is continuous but not differentiable at $$x = 1$$? 4. How does the choice of coefficients $$c$$ and $$d$$ influence the graph of the function near $$x = 1$$? 5. What would the graph of $$f(x)$$ look like if $$c$$ or $$d$$ had different values? **Tip:** When solving for differentiability of piecewise functions, always check both continuity and derivative equality at the boundary points!

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Math Problem Analysis

Mathematical Concepts

Differentiability
Piecewise Functions
Continuity

Formulas

f(x) = {x + 1 if x < 1, x^2 + cx + d - c if x >= 1}
f'(x) = 1
f'(x) = 2x + c

Theorems

Conditions for Differentiability
Continuity at a Point

Suitable Grade Level

Grades 11-12 or Early College