Question

Let \(\mathrm{f}, \mathrm{g}\) and \(\mathrm{h}\) be real-valued functions defined on the interval \([0,1]\) by \(\mathrm{f}(\mathrm{x})=\mathrm{e}^{\mathrm{x}^{2}}+\mathrm{e}^{-\mathrm{x}^{2}}, \mathrm{~g}(\mathrm{x})=\mathrm{x} \mathrm{e}^{\mathrm{x}^{2}}+\mathrm{e}^{-\mathrm{x}^{2}}\) and \(\mathrm{h}(\mathrm{x})=\mathrm{x}^{2} \mathrm{e}^{\mathrm{x}^{2}}+\mathrm{e}^{-\mathrm{x}^{2}}\). If \(\mathrm{a}, \mathrm{b}\) and \(\mathrm{c}\) denote, respectively, the absolute maximum of \(\mathrm{f}, \mathrm{g}\) and \(\mathrm{h}\) on \([0,1]\), then A) \(a=b\) and \(c \neq b\) B) \(a=c\) and \(a \neq b\) C) \(a \neq b\) and \(c \neq b\) D) \(a=b=c\)

Short answer

a=b=c, hence the answer is D) a=b=c.

Step-by-step solution

Understand the Nature of the Functions

Recognize that the functions are continuous on the closed interval [0,1], thus the Extreme Value Theorem guarantees that each function will attain an absolute maximum and minimum on the interval.

Find Maximum of f(x)

Since f(x) is always positive and increasing on [0,1], its maximum is at the endpoint x=1, yielding f(1)=e+1/e.

Find Maximum of g(x)

The maximum of g(x) occurs either at an endpoint or at a critical point where g'(x)=0. Compute the derivative of g, set it to zero, and check the value of g at x=0 and x=1. If no critical points exist in [0,1], the maximum is at one of the endpoints. Since g(x) is increasing, the maximum value of g is at x=1, that is, g(1)=e+1/e.

Find Maximum of h(x)

Similar to g(x), the maximum of h(x) is either at an endpoint or at a critical point. Compute the derivative, set it equal to zero, and find the value at x=0 and x=1. Since h(x) has the same increasing nature, its maximum is also at x=1, yielding h(1)=e+1/e.

Evaluate and Compare Maximum Values

The absolute maxima of f, g, and h are all the same when evaluated at x=1. Therefore, a=b=c.

Key concepts

Extreme Value Theorem

The Extreme Value Theorem (EVT) is a fundamental result in calculus which states that a continuous function defined on a closed interval must have both an absolute maximum and an absolute minimum on that interval. This theorem is incredibly important in the study of calculus because it assures us that, under certain conditions, a function will reach its highest and lowest points.

For example, in the given exercise, the functions \(f(x)\), \(g(x)\), and \(h(x)\) are all continuous on the closed interval [0,1], and therefore, according to the EVT, each of them will have an absolute maximum value within this interval. Knowing this allows us to restrict our search for maximum values to this interval and guarantees that the search will be successful.

Maxima and Minima

Maxima and minima refer to the highest and lowest values of a function, respectively. A function can have 'local' maxima and minima, which are the highest or lowest points within a surrounding neighborhood of a point, and 'absolute' or 'global' maxima and minima, which are the highest or lowest points on the entire domain of the function.

In the context of the exercise provided, the focus is on finding the absolute maxima of the functions \(f\), \(g\), and \(h\) on the closed interval [0,1]. The absolute maximum is where the function reaches its highest value over this specific domain, and it can occur at critical points where the derivative equals zero or does not exist, or at the endpoints of the interval. Identifying this maximum value is crucial for solving many problems in calculus and applied mathematics, such as optimization.

Differentiation and Critical Points

Differentiation is a core concept in calculus that deals with how a function changes at any given point. It is the process of finding the derivative of a function, which provides the rate of change or the slope of the function at any point. Critical points of a function occur where the derivative is zero or undefined. These points are important because they are possible locations for local maxima or minima.

In our exercise, we differentiate functions \(g(x)\) and \(h(x)\) to find their critical points. To do this, we calculate the derivatives \(g'(x)\) and \(h'(x)\), and set them equal to zero. We then evaluate the functions at these critical points, as well as at the endpoints of the interval [0,1], to determine where the functions achieve their maximum values. It is also important to compare these values to decide if any of the maxima are equal, as in the case of the exercise's conclusion that \(a=b=c\).