Math Problem Statement

Determine the intervals on which the following function is concave up or concave down. Identify any inflection points. f left parenthesis x right parenthesisequalsStartRoot 4 x EndRoot ln left parenthesis x right parenthesis Question content area bottom Part 1 Determine the intervals on which the following functions are concave up or concave down. Select the correct choice​ and, if​ necessary, fill in the answer​ box(es) to complete your choice. A. The function is concave up on    enter your response here and concave down on    enter your response here. ​(Simplify your answers. Type your answers in interval notation. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the​ expression.) B. The function is concave down on    enter your response here and the function is never concave up. ​(Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the​ expression.) C. The function is concave up on    enter your response here and the function is never concave down. ​(Simplify your answer. Type your answer in interval notation. Use a comma to separate answers as needed. Use integers or fractions for any numbers in the​ expression.) D. The function is never concave up nor concave down. Part 2 Locate any inflection points of f. Select the correct choice​ and, if​ necessary, fill in the answer box to complete your choice. A. An inflection point occurs at xequals    enter your response here. ​(Type an integer or a simplified fraction. Use a comma to separate answers as​ needed.) B. There are no inflection points for f.

Solution

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Math Problem Analysis

Mathematical Concepts

Differentiation
Concavity
Inflection Points
Product Rule
Quotient Rule
Logarithmic Functions
Square Root Functions

Formulas

First derivative: f'(x) = (ln(x) + 1)/√x
Second derivative: f''(x) = (1/x)√x - (ln(x) + 1)/(2√x)

Theorems

Product Rule of Differentiation
Quotient Rule of Differentiation

Suitable Grade Level

Grades 11-12