f’(x) = 0

Therefore

Question

f’(x) = 0

Therefore

BlackTom AI · Accepted Answer

When f'(x) = 0, we are locating points where the slope of the tangent is horizontal, which identifies critical points on the function's graph.

Option 1: 'f(x) is at a minimum or a maximum' captures the standard consequence of encountering a critical point. At these x-values, the function can have a local maximum, a local minimum, or sometimes a horizontal inflection point depending on higher-order behavior. Thus this statement aligns with the general theory of critical points and extrema.

Option 2: 'f(0) is at a minimum' asserts something specific about the function value at x = 0. The condition f'(x) = 0 does not imply x = 0 is a special or minimum point unless additional information about the function is given. Therefore this claim is not warranted from f'(x) = 0 alone.

Option 3: 'f(x) is at a maximum' claims a maximum occurs universally at the critical point. In reality, a critical point can be a maximum, a minimum, or neither (e.g., a flat point or inflection with horizontal tangent). Ruling out minima and inflections makes this statement too restrictive.

Option 4: 'f(x) is at a minimum' faces the same issue as Option 3 but in the opposite direction: not every critical point yields a minimum. Without further test (second derivative test or sign-change analysis), we cannot claim a minimum solely from f'(x) = 0.

In summary, the most general and accurate takeaway from f'(x) = 0 is that the function has potential extrema at points where the derivative vanishes, i.e., where 'f(x) is at a minimum or a maximum' may occur. The other statements impose unwarranted specifics about the exact type or location of the extremum.

ERMCPS5350_003_2025_3 - INTRO TO QUANTITATIVE RISK MANAGEMENT 10. Quiz for Class 10

f’(x) = 0 Therefore

View Explanation

Log in for full answers

Similar Questions

Determine all critical points for the function. f(x) = (x - 2)7

Question at position 2 The function f(x,y)=13x3+12y2+xy−6x+3 has a relative minimum at(-2, 2)(-3, 3)(2, -2)(3, -3)(2, 2)

Question at position 1 The number of critical points of f(x,y)=x2+x2y+y2−2y+2 is31240

Question at position 2 The function f(x,y)=13x3+12y2+xy−6x+3f\left(x,y\right)=\frac{1}{3}x^3+\frac{1}{2}y^2+xy-6x+3 has a relative minimum at(-2, 2)(2, -2)(-3, 3)(2, 2)(3, -3)

Question at position 1 The number of critical points of f(x,y)=x2+x2y+y2−2y+2f\left(x,y\right)=x^2+x^2y+y^2-2y+2 is42301

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler