Math Problem Statement
Let \( f(x) = \begin{cases} x^2 - 6, & x < 0 \\ -6, & x \geq 0 \end{cases} \). (A) Sketch the graph of \( f \). (B) Find the value of \( x \) where \( f \) is discontinuous. (C) Find the value of \( x \) where \( f \) is not differentiable.
Solution
Let's analyze the function given in the image:
x^2 - 6, & x < 0 \\ -6, & x \geq 0 \end{cases}$$ ### Part (A): Sketch the Graph The function has two parts: 1. For $$x < 0$$, $$f(x) = x^2 - 6$$, which is a parabola shifted down by 6 units. 2. For $$x \geq 0$$, $$f(x) = -6$$, which is a constant function. To sketch: - For $$x < 0$$, the parabola $$f(x) = x^2 - 6$$ opens upwards, and its vertex is at $$(0, -6)$$. - For $$x \geq 0$$, the graph is a horizontal line at $$f(x) = -6$$. ### Part (B): Continuity Check To check for continuity, observe what happens at $$x = 0$$: 1. The left-hand limit as $$x \to 0^-$$ is $$\lim_{x \to 0^-} f(x) = 0^2 - 6 = -6$$. 2. The right-hand limit as $$x \to 0^+$$ is $$\lim_{x \to 0^+} f(x) = -6$$. Since the left-hand limit, right-hand limit, and the function value at $$x = 0$$ all equal $$-6$$, the function is **continuous** at $$x = 0$$. Therefore, **no points of discontinuity exist** for $$f(x)$$. Answer for Part (B): **NONE** ### Part (C): Differentiability Check To check differentiability at $$x = 0$$: 1. Compute the derivative for $$x < 0$$: $$f'(x) = 2x$$. - As $$x \to 0^-$$, $$f'(x) \to 0$$. 2. For $$x \geq 0$$, $$f(x) = -6$$, so $$f'(x) = 0$$ for $$x > 0$$. However, at $$x = 0$$, there is a **cusp** due to the abrupt change from a quadratic function to a constant function. Hence, $$f$$ is **not differentiable** at $$x = 0$$. Answer for Part (C): $$x = 0$$ Would you like further details on any part? --- ### Follow-Up Questions: 1. How is differentiability related to continuity at a point? 2. What are common indicators of a function being non-differentiable at a point? 3. Why does a cusp cause non-differentiability? 4. How can we verify continuity for piecewise functions in general? 5. What other examples of functions have similar behavior to this one at a boundary? ### Tip: For piecewise functions, checking continuity and differentiability at transition points is essential to understand their behavior fully.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Piecewise functions
Continuity
Differentiability
Formulas
Limit definition of continuity
Derivative rules for polynomials
Theorems
Continuity at a point
Differentiability implies continuity
Suitable Grade Level
Grades 11-12
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