Find the exact location of all the relative and absolute extrema of the function with domain $( - \infty , \infty )$ .
$f ( x ) = e ^ { - 5 x ^ { 2 } }$
A) $( 0,1 )$ - absolute maximum
B) $( 1,5 )$ - absolute maximum
C) $( 0,2 )$ - relative minimum
D) $( 1,0 )$ - absolute maximum
E) $( 0,1 )$ - absolute minimum

Question

Find the exact location of all the relative and absolute extrema of the function with domain $( - \infty , \infty )$ . 
  $f ( x ) = e ^ { - 5 x ^ { 2 } }$ 
A) $( 0,1 )$ - absolute maximum
B) $( 1,5 )$ - absolute maximum
C) $( 0,2 )$ - relative minimum
D) $( 1,0 )$ - absolute maximum
E) $( 0,1 )$ - absolute minimum

Quizplus · Accepted Answer

The function $f(x) = e^{-5x^2}$ is a Gaussian function, which is symmetric about the y-axis and has its maximum value when $x=0$. The value of the function at $x=0$ is $e^0 = 1$, making $(0,1)$ the point of absolute maximum. There are no relative minima or other extrema because the function continuously decreases as $x$ moves away from 0 in both directions, and it approaches 0 but never reaches a minimum value greater than 0.

Find the Exact Location of All the Relative and Absolute (−∞,∞)( - \infty , \infty )(−∞,∞)

Find the Exact Location of All the Relative and Absolute $( - \infty , \infty )$