Here are a couple questions from the old official SAT Subject Test Math I practice exam:

The function f is defined by f(x) is x^4 – 4x^2 + x + 1 for -5 ≤ x ≤ 5. In which of the following intervals does the minimum value of f occur?
A) -5 ≤ x ≤ -3
B) -3 ≤ x ≤ -1
C) -1 ≤ x ≤ 1
D) 1 ≤ x ≤ 3
E) 3 ≤ x ≤ 5

Can you solve w/o graphing?

Yes, but just before beginning I think it’s important to stress HOW USEFUL a graphing calculator is for the math Subject Tests. Anyone who’s prepping for those that doesn’t own one and can’t borrow one from school should find another way to get their hands on one they know how to use for test day.

To get this without graphing, you’re going to want to plug in values Just the integers should work. Note that x^4 will always be 0 or positive and -4x^2 will always be 0 or negative, but x will be both positive and negative, you should start by plugging in negative numbers.

(-5)^4-4(-5)^2-5+1=521

(-4)^4-4(-4)^2-4+1=189

(-3)^4-4(-3)^2-3+1=43

(-2)^4-4(-2)^2-2+1=-1

(-1)^4-4(-1)^2-1+1=-3

(0)^4-4(0)^2-0+1=-1

From there, you probably see that the minimum is around –1. So plug in a couple more values to see whether you want to choose B or C.

(-1.5)^4-4(-1.5)^2-1.5+1=-4.4375

(-0.5)^4-4(-0.5)^2-0.5+1=-0.4375

Yeah…gonna want to go with B.

Again, the graph is SO helpful here:

With that, you immediately see that the minimum is between –1 and –2.