Home › Functions and graphs

Find the minimum of an exponential with a quadratic exponent

For a base above one, minimize the exponent

The function $b^{t}$ increases with $t$ when $b > 1$ . Therefore, to minimize $b^{f (x)}$ , first find the minimum of $f (x)$ .

Problem

Find the minimum value of $y = 4^{x^{2} - 6 x + 12}$ over all real $x$ .

The minimum value is

64

, attained at

x = 3

The quadratic exponent, not the base 4, determines where the minimum occurs.

Rewrite the exponent as $x^{2} - 6 x + 12 = (x - 3)^{2} + 3$ . Since a square is nonnegative, the exponent is at least 3, with equality at $x = 3$ .

Because the base $4$ is greater than 1, $4^{t}$ increases as $t$ increases. The smallest possible value is therefore $4^{3} = 64$ .

Screenshot any problem on your screen and get the steps. Chrome on a computer; free.