Question
upstudy study bank question image url

Example 14 Find and classify the critical points of \( f(x)=x^{4}-4 x^{3}+5 \).

Ask by Gardner Dickson. in Myanmar
Mar 17,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The critical points are at \( x=3 \) (local minimum) and \( x=0 \) (stationary inflection point).

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

The Deep Dive

To find the critical points of the function \( f(x) = x^4 - 4x^3 + 5 \), you first need to take the derivative of the function: \[ f'(x) = 4x^3 - 12x^2. \] Next, set the derivative equal to zero to find the critical points: \[ 4x^3 - 12x^2 = 0 \implies 4x^2(x - 3) = 0. \] This gives us the solutions \( x = 0 \) and \( x = 3 \). Now, to classify these critical points, we'll evaluate the second derivative: \[ f''(x) = 12x^2 - 24x. \] Now, substitute the critical points into the second derivative: For \( x = 0 \): \[ f''(0) = 12(0^2) - 24(0) = 0. \] For \( x = 3 \): \[ f''(3) = 12(3^2) - 24(3) = 108 - 72 = 36. \] Since \( f''(3) > 0 \), the point \( x = 3 \) is a local minimum. The second derivative test is inconclusive for \( x = 0 \) since \( f''(0) = 0 \). Thus, we can further analyze by checking the first derivative around \( x = 0 \): 1. For \( x < 0 \), \( f'(x) < 0 \) (decreasing). 2. For \( x > 0 \), \( f'(x) > 0 \) (increasing). Thus, \( x = 0 \) is a local maximum. In summary, the critical points are \( x = 0 \) (local maximum) and \( x = 3 \) (local minimum).

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy