If the equation ^4+^4+=0 has real solutions for , then lies in the interval:

Question

If the equation $\cos^4	heta+\sin^4	heta+\lambda=0$ has real solutions for $	heta$, then $\lambda$ lies in the interval:

Accepted Answer

$$\lambda = -[\cos^4\theta+\sin^4\theta] = -\left[1-2\sin^2\theta\cos^2\theta\right] = \frac12\sin^2(2\theta) - 1$$

Since $\sin^2(2\theta)\in[0,1]$: $\tfrac12\sin^2(2\theta)\in\left[0,\tfrac12\right]$, so:

$$\lambda \in \left[-1,-\frac12\right]$$

Solutions · 1