Math Problem Statement

Bestem k slik at f blir kontinuerlig. Begrunn at det ikke fins en k som kan gjøre f deriverbar.

Solution

To solve the problem presented in the image, let's analyze the given information:

The function f(x)f(x) is defined piecewise as:

\begin{cases} 3e^{2x}, & x < k \\ 2e^{3x}, & x \geq k \end{cases}$$ ### Part (a) Determine $$k$$ such that $$f$$ is continuous. For $$f(x)$$ to be continuous, the two pieces of $$f(x)$$ must meet at $$x = k$$. This requires: $$\lim_{x \to k^-} f(x) = \lim_{x \to k^+} f(x) = f(k).$$ 1. **Left-hand limit (as $$x \to k^-$$):** $$\lim_{x \to k^-} f(x) = 3e^{2k}.$$ 2. **Right-hand limit (as $$x \to k^+$$):** $$\lim_{x \to k^+} f(x) = 2e^{3k}.$$ For continuity, these limits must be equal: $$3e^{2k} = 2e^{3k}.$$ Divide through by $$e^{2k}$$ (valid since $$e^{2k} > 0$$): $$3 = 2e^{k}.$$ Solve for $$k$$: $$e^{k} = \frac{3}{2}.$$ Taking the natural logarithm: $$k = \ln\left(\frac{3}{2}\right).$$ Thus, $$k = \ln\left(\frac{3}{2}\right)$$ ensures $$f(x)$$ is continuous. --- ### Part (b) Show that no $$k$$ can make $$f$$ differentiable. For $$f(x)$$ to be differentiable at $$x = k$$, the derivatives of the two pieces must also match at $$x = k$$. Compute the derivatives of each piece: 1. **Derivative for $$x < k$$:** $$f'(x) = \frac{d}{dx}[3e^{2x}] = 6e^{2x}.$$ At $$x = k$$, this becomes: $$f'(k^-) = 6e^{2k}.$$ 2. **Derivative for $$x \geq k$$:** $$f'(x) = \frac{d}{dx}[2e^{3x}] = 6e^{3x}.$$ At $$x = k$$, this becomes: $$f'(k^+) = 6e^{3k}.$$ For $$f(x)$$ to be differentiable at $$x = k$$, we must have: $$f'(k^-) = f'(k^+).$$ This requires: $$6e^{2k} = 6e^{3k}.$$ Simplify by dividing through by $$6e^{2k}$$: $$1 = e^{k}.$$ Taking the natural logarithm: $$k = 0.$$ However, if $$k = 0$$, substituting into the continuity condition: $$3e^{2k} = 2e^{3k} \implies 3 = 2,$$ which is a contradiction. Thus, no value of $$k$$ can make $$f(x)$$ differentiable. --- ### Final Answers: (a) $$k = \ln\left(\frac{3}{2}\right)$$ ensures $$f(x)$$ is continuous. (b) No $$k$$ can make $$f(x)$$ differentiable. --- Would you like a detailed explanation of any step, or additional examples? ### Related Questions: 1. What does continuity imply about the relationship between limits and function values? 2. How does differentiability depend on continuity? 3. Can a piecewise function ever be differentiable at a boundary point? 4. What happens when the slopes of two pieces of a function do not match at a point? 5. How would the analysis change if the function were defined differently? ### Tip: Always verify continuity before checking differentiability; differentiability implies continuity, but not vice versa!

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Math Problem Analysis

Mathematical Concepts

Continuity of functions
Differentiability of piecewise functions
Exponential functions

Formulas

Continuity condition: lim(x→k⁻) f(x) = lim(x→k⁺) f(x) = f(k)
Differentiability condition: f'(k⁻) = f'(k⁺)
Exponential rules: e^(a+b) = e^a * e^b

Theorems

A function is differentiable at a point only if it is continuous at that point.
Limits and continuity principles.

Suitable Grade Level

High School (Grades 11-12) or University Introductory Level