Find the maximum of y=x^a-x^b on [0,1], where 0 < a < b . In particular, find the maximum of y=x

Find the maximum of y=x^a-x^b on [0,1], where 0 < a

Key Concepts

Power Functions and Exponent Properties

Power functions are functions of the form x^n, where n is a constant exponent. Understanding their behavior, especially on the interval [0,1], is important because they often exhibit different characteristics based on the exponent's value. When optimizing expressions that involve differences of power functions, it is essential to manipulate these exponents to simplify and solve equations for extrema.

Optimization in Closed Intervals

This concept involves determining the maximum or minimum values of a continuous function defined on a closed interval. The extreme value theorem guarantees that a continuous function on a closed interval achieves both a maximum and a minimum value, so it is necessary to check both the critical points, where the derivative is zero or undefined, and the boundary points of the interval.

Differentiation and Critical Points

Differentiation is a fundamental tool in calculus used to find the rate of change of a function. By setting the derivative equal to zero, one can identify the critical points where the function's slope is zero, indicating potential local maxima or minima. This technique is especially useful for polynomial or power functions where the power rule applies.

Transcript

00:01 Number 81 we need to find a maximum value of this function between 0 and 1 where a is less than b but is positive so let's find the critical points first so we have d y over d x using the chain rule we have a x a minus 1 and we have bx b minus 1 this can be read written as a x raised to the power a over x minus b x raised to the power b over x this is y over d x to find the critical points, we'll equate that to zero.

00:30 So we have d -y over d -x is equated to zero.

00:34 This means that we have a -x -raised to the power a minus b -x -raised to the power b over x is equated to zero.

00:42 This means that a -x -raised -to -the -power -a -minus b -x -reased -to -the -power -b is equal to 0.

00:48 It means that we have a -x -raised -to -the -power -b.

00:53 If we rearrange this, we get x -raise -to -the -power -a over x.

00:57 X raised to the power b is b over a so this comes out as x can be written as x it just can be written as x raised to the power a minus b because the powers uh because when we take it up the power will go negative so x raised to the power a minus b is b over a if we divide both sides the power of both sides by one over a minus b then x is equal to b over a raised to the power one over a minus b this is the value of x for which we have a critical point and let's see if we have a local maxima or local minima here if we use the double derivative test so this is our derivative so let's differentiate this once again so we have d square y over d x a square we got to use the quotient rules so we have denominator square we have denominator as it is differentiation of numerator would be a as a in fact this space won't be enough so let's do it separately so we have uh in fact we can skip testing whether this has a local maxima or minima since this is just one point and if we get the endpoints behavior we can uh show you find out how it behaves.

02:32 So endpoint is 0...

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