Chapter 3: Q. 82 (page 311)

determine the global extrema of function $f (x) = x^{2} \ln (0 \cdot 2 x), I = (0, 4), J = (0, \infty)$

Short Answer

Expert verified

On I, f has a global maximum at x=0 and a global minimum at $x = \frac{5}{\sqrt{e}}$

On J, f has no global minimum at $x = \frac{5}{\sqrt{e}}$ and has no global maximum

Step by step solution

Step 1. Given information

The given function is $f (x) = x^{2} \ln (0 \cdot 2 x)$

Step 2. Calculate global maximum and minimum

$f (x) = x^{2} \ln (0 \cdot 2 x) f' (x) = 0 \frac{d (x^{2} \ln (0 \cdot 2 x))}{d x} = 0 x^{2} \cdot \frac{1}{0 \cdot 2 x} \times 0 \cdot 2 + \ln (0 \cdot 2 x) \times 2 x = 0 x (1 + 2 \ln (0 \cdot 2 x)) = 0 x = 0; x = \frac{e^{\frac{- 1}{2}}}{0 \cdot 2} x = 0; x = \frac{5}{\sqrt{e}} A g a i n, \lim_{x \to 0} f (x) = \lim_{x \to 0} x^{2} \ln (0 \cdot 2 x) = \lim_{x \to 0} \frac{\frac{1}{0 \cdot 2 x} \times 0 \cdot 2}{\frac{- 2}{x^{3}}} = \lim_{x \to 0} (\frac{- 1}{x} \cdot \frac{x^{3}}{2}) = \lim_{x \to 0} \frac{- x^{2}}{2} A g a i n \lim_{x \to \infty} f (x) = \lim_{x \to \infty} x^{2} \ln (0 \cdot 2 x) = \infty$

Step 3. The solution

On I, f has a global maximum at x=0 and a global minimum at $x = \frac{5}{\sqrt{e}}$

On J, f has no global minimum at $x = \frac{5}{\sqrt{e}}$ and has no global maximum

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determine the global extrema of function $f (x) = x^{2} \ln (0 \cdot 2 x), I = (0, 4), J = (0, \infty)$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculate global maximum and minimum

Step 3. The solution

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determine the global extrema of functionfx=x2ln0·2x,I=0,4,J=0,∞

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculate global maximum and minimum

Step 3. The solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

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Logic and Functions

Study anywhere. Anytime. Across all devices.

determine the global extrema of function $f (x) = x^{2} \ln (0 \cdot 2 x), I = (0, 4), J = (0, \infty)$