Question 274·Hard·Math

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The functions below are defined for $x \ge 0$ .

f(x)=20(0.3)^{x^{2}+1}

II.

g(x)=20(0.09)(0.3)^{x^{2}-2}

Which of the following equations displays, as a constant or coefficient, the maximum value of the function it defines, where $x \ge 0$ ?

A base between $0$ and $1$ signals exponential decay: the power gets smaller as its exponent grows. On a restricted domain, find where each exponent is smallest, then evaluate the functions there in Desmos. Compare those maximum outputs with the numbers displayed in each equation. A number in front isn't automatically the maximum when the exponent has a constant added or subtracted.

Hints

Hint 1
For a base between $0$ and $1$ , a larger exponent gives a smaller power. Since $x\ge0$ , where is $x^2$ smallest? Both given exponents change with $x^2$ .
Hint 2
A function's maximum is its greatest output over the allowed inputs, not necessarily the number in front. Evaluate each formula at the input you found. What does the $+1$ in I's exponent do there?
Hint 3
A negative exponent means a reciprocal, not a negative output. At $x=0$ , II's exponent is $-2$ . How does that power interact with $0.09=(0.3)^2$ ?

Step-by-step

Find each maximum at the allowed endpoint
Step 1
Find the input that gives the largest outputs
Because $x\ge0$ , the smallest $x^2$ can be is $0$ , at $x=0$ . Both given exponents increase with $x^2$ , and both factors in front of the powers are positive. For a base between $0$ and $1$ , the smallest exponent gives the largest output. So both maxima occur at $x=0$ .
Step 2
Evaluate I at its maximum
Type $f(x)=20(0.3)^{x^2+1}$ , then $f(0)$ . Desmos prints $6$ . At $x=0$ , I's exponent is $1$ , not $0$ , so its displayed $20$ is not its maximum.
Step 3
Evaluate II and compare its displayed factor
Add $g(x)=20(0.09)(0.3)^{x^2-2}$ , then $g(0)$ . Desmos prints $20$ . At $x=0$ , the exponent $-2$ makes a reciprocal of $(0.3)^2$ ; that cancels $0.09=(0.3)^2$ . So II's displayed factor $20$ is its maximum, while I's maximum is $6$ . Choice B.

Question 274·Hard·Math

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