Find the absolute minimum value of the function g(x)=exx,x&gt;0

Step 1 The function is g(x)=exx for x&gt;0. Differentiate the function with respect to x. ddxg(x)=ddx(exx) =xex−exe2x =x−1ex Equate the derivative to 0 to find the critical points. ddxg(x)=0 x−1ex=0 x-1=0 x=1 Step 2 Determine the second derivative of the function. ddx(ddxg(x))=ddx(x−1ex) =ex−(x−1)exe2x =1−(x−1)ex =2−xex Substitute 1 for x in the above equation. ddx(ddxg(1))=2−1e1 =1e Since, the second derivative is positive, the point x = 1 is the point of minima. Step 3 Substitute 1 for x in the equation g(x)=exx. g(1)=e11 =e Thus, the absolute minimum value of the function is e at x = 1.

Expert Answer to "Find the absolute minimum value of the function..."

High School
Algebra
Algebra I
Piecewise-Defined Functions

hionormf

Answered question

2021-12-09

Find the absolute minimum value of the function

g (x) = \frac{e^{x}}{x}, x > 0

Answer & Explanation

Ben Owens

Beginner2021-12-10Added 27 answers

Step 1
The function is

g (x) = \frac{e^{x}}{x}

for x>0.
Differentiate the function with respect to x.

\frac{d}{d x} g (x) = \frac{d}{d x} (\frac{e^{x}}{x})

= \frac{x e^{x} - e^{x}}{e^{2 x}}

= \frac{x - 1}{e^{x}}

Equate the derivative to 0 to find the critical points.

\frac{d}{d x} g (x) = 0

\frac{x - 1}{e^{x}} = 0

x-1=0
x=1
Step 2
Determine the second derivative of the function.

\frac{d}{d x} (\frac{d}{d x} g (x)) = \frac{d}{d x} (\frac{x - 1}{e^{x}})

= \frac{e^{x} - (x - 1) e^{x}}{e^{2 x}}

= \frac{1 - (x - 1)}{e^{x}}

= \frac{2 - x}{e^{x}}

Substitute 1 for x in the above equation.

\frac{d}{d x} (\frac{d}{d x} g (1)) = \frac{2 - 1}{e^{1}}

= \frac{1}{e}

Since, the second derivative is positive, the point x = 1 is the point of minima.
Step 3
Substitute 1 for x in the equation

g (x) = \frac{e^{x}}{x}

g (1) = \frac{e^{1}}{1}

=e
Thus, the absolute minimum value of the function is e at x = 1.

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