Find the absolute maximum and absolute minimum values of f on the given interval. f(x)=2x3−3x2−36x+5,[−3,4] absolute minimum value absolute maximum value

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Find the absolute maximum and absolute minimum values of f on the given interval.  f(x)=2x3−3x2−36x+5,[−3,4]  absolute minimum value  absolute maximum value

zesponderyd · Accepted Answer

Step 1Given: f(x)=2x3−3x2−36x+5,[−3,4]If f&quot;(c)&amp;lt;0, then maximum value of a function exist at x=c, where x=c is the critical pointIf f&quot;(c)&amp;gt;0, then minimum value of a function exist at x=c, where x=c is the critical point.Critical point of the function are the points where f&#039;(x)=0Step 2Therefore, f′(x)=6x2−6x−36To find the critical pointf′(x)=6x2−6x−36=0⇒x2−x−6=0⇒x2−3x+2x−6=0⇒x(x−3)+2(x−3)=0⇒(x+2)(x−3)=0⇒x=−2,x=3Step 3Now, f&#039;&#039;(x)=12x-6f&#039;&#039;(-2)=12(-2)-6=-30⇒f″(−2)&amp;lt;0So, maximum exist at x=−2Maximum value of f(−2)=2(−2)3−3(−2)2−36(−2)+5=49andf&#039;&#039;(3)=12(3)-6=30⇒f″(3)&amp;gt;0So, minimum exist at x=3Minimum value of f(3)=2(3)3−3(3)2−36(3)+5=−76

Find the absolute maximum and absolute minimum values of f

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