Find the critical points of the following function on the given interval. Identify the absolute maximum and absolute minimum values (if they exist). g(x)=x4−50x2 on [-1,5]

Question

Find the critical points of the following function on the given interval. Identify the absolute maximum and absolute minimum values (if they exist).  g(x)=x4−50x2 on [-1,5]

Gerald Lopez · Accepted Answer

Given,g(x)=x4−50x2,[−1,5]Differentiating with respect to x, we getg′(x)=4x3−50(2x)   [∵d(xn)dx=n⋅xn−1]=4x3−100xNow critical values are the values of x for which first derivative is zero.Therefore solve the first derivative by equating it to zero. That is,g&#039;(x)=0⇒4x3−100x=0⇒4x(x2−25)=0⇒x=0 or x=−5 or x=5Since −5 not ∈[−1,5], therefore the critical values on the given interval are x=0 &amp;amp; x=5.Step 2Now we find the value of the given function at these critical values and at the end points of the given interval.When x=-1,g(−1)=(−1)4−50(−1)2=-49When x=0,g(0)=(0)4−50(0)2=0When x=5,g(5)=(5)4−50(5)2=625-1250=-625Hence absolute maximum value is 0 occurs at x=0 and absolute minimum value is -625 occurs at x=5.

Find the critical points of the following function on the

Answered question

Answer & Explanation

New Questions in Algebra I