Find the absolute maximum and absolute minimum values of f(x)=x2−25x2+25 on the interval [-5,5].[-5,5]. Find the absolute maximum of ff on the interval. Find the absolute minimum of ff on the interval.

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Find the absolute maximum and absolute minimum values of f(x)=x2−25x2+25 on the interval [-5,5].[-5,5].  Find the absolute maximum of ff on the interval.  Find the absolute minimum of ff on the interval.

David Clayton · Accepted Answer

Step 1Given,f(x)=x2−25x2+25 and the interval [-5,5].f(x) can be written as, f(x)=−24x2+25Step 2Absolute maximum or absolute minimum values of a function lies at end points of the interval or at the points where first derivative is zero.So first we find the derivative of the given function, that isf′(x)=−24(2x)+0   [∵ddx(xn)=n⋅xn−1 &amp;amp; ddx(c)=0 for c∈R]=-48xStep 3Now finding the points where first derivative is zero, that isf&#039;(x)=0⇒−48x=0⇒x=0Step 4Now computing the value of the given function at the end points of the given interval and at x =0.We getf(−5)=(−5)2−25(−5)2+25=−575f(5)=(5)2−25(5)2+25=−575f(0)=(0)2−25(0)2+25=25Step 5Hence,absolute maximum value of f is 25 occurs at the point x=0.absolute minimum value of f is -575 occurs at the points x=-5 and x=5.

Find the absolute maximum and absolute minimum values of f(x)=x^{2}-25x^{2}+25

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