The original question includes a diagram: Four graphs P, Q, R and S on 0 < x < pi/2: P falls from 1 to 0, Q rises from 0 to 1, and R and S each start and end at 1 with a dip in between.
Four graphs P, Q, R and S show y = (cos x)cos x, y = (sin x)sin x, y = (cos x)sin x, y = (sin x)cos x for 0 < x < π2 in some order. Graph P falls from 1 to 0; graph Q rises from 0 to 1; graphs R and S both begin and end at 1, dipping in between, with R's dip later than S's.
Which row correctly identifies the graphs, in the order (cos x)cos x, (sin x)sin x, (cos x)sin x, (sin x)cos x?
- AP, Q, R, S
- BP, Q, S, R
- CQ, P, R, S
- DQ, P, S, R
- ER, S, P, Q
- FR, S, Q, P
- GS, R, P, Q
- HS, R, Q, P
Show the answer and worked solution
answer · E
- AP, Q, R, S
- BP, Q, S, R
- CQ, P, R, S
- DQ, P, S, R
- ER, S, P, Q
- FR, S, Q, P
- GS, R, P, Q
- HS, R, Q, P
Check the two ends. As x → 0, cos x → 1 and sin x → 0, so (cos x)sin x → 1 while (sin x)cos x → 0; as x → π2 these swap. So the falling graph P is (cos x)sin x and the rising graph Q is (sin x)cos x. The other two both tend to 1 at each end, since tt → 1 as t → 0+ and at t = 1. Writing g(t) = tt, its minimum is at t = 1e: for (sin x)sin x that happens at x ≈ 0.38, early in the range, and for (cos x)cos x at x ≈ 1.19, late. So R is (cos x)cos x and S is (sin x)sin x.