User Name Remember Me? Password

 Calculus Calculus Math Forum

 May 9th, 2014, 07:27 PM #1 Senior Member   Joined: Feb 2014 From: Louisiana Posts: 156 Thanks: 6 Math Focus: algebra and the calculus Minima What is the minimum value of the function $\displaystyle f(x) = 2ax^3 - 9ax^2 + 5a$ on the closed interval $\displaystyle [-1, 2]$? State all of the possible minimums in terms of how $\displaystyle a$ varies. May 9th, 2014, 08:06 PM #2 Global Moderator   Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond Differentiate f(x) once. There is an extremum at x = 0 which is a minimum if a is negative. The minimum of the function on [-1, 2] is then 5a. May 9th, 2014, 08:13 PM #3 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,691 Thanks: 2670 Math Focus: Mainly analysis and algebra \begin{align*} f(x) &= 2ax^3 - 9ax^2 + 5a \\ f'(x) &= 6ax^2 - 18ax \\ \end{align*} So at a turning point: \begin{align*} 6ax^2 &= 18ax \\ x &= \begin{cases}-3 &\text{or} \\ 0 \\ \end{cases} \\ \end{align*} So, when $a \gt 0$, we have a minimum at $x = 0$. When $a \lt 0$, we have only a local maximum in [-1, 2], so the minimum value must be at one of the endpoints. $$f(-1) = -2a -9a + 5a = -6a \\ f(2) = 16a - 36a + 5a = -15a \\ f(-1) \lt f(2)$$ since $a \lt 0$. Thanks from Mr Davis 97 May 9th, 2014, 08:42 PM   #4
Global Moderator

Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,968
Thanks: 1152

Math Focus: Elementary mathematics and beyond
Quote:
 Originally Posted by v8archie So, when $a \gt 0$, we have a minimum at $x = 0$.
I disagree. Try graphing it. May 9th, 2014, 09:15 PM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,691 Thanks: 2670 Math Focus: Mainly analysis and algebra Oops! I got a sign error on the other root. Never mind, the method is clear enough. Tags maxima, minima Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post helloprajna Calculus 4 October 13th, 2012 05:42 AM helloprajna Calculus 3 October 13th, 2012 04:03 AM kingkos Calculus 3 April 10th, 2012 08:20 AM ArmiAldi Real Analysis 1 March 6th, 2008 04:51 AM bhuvi Algebra 1 December 31st, 1969 04:00 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top       