Math Help - Extrema on interval?

1. Extrema on interval?

Find the critical numbers of the function. I have two(?).

1. x square root(4-x) (This is what I did)

square root(4-x)(1)+(x) ((4-x))/((2 square root(4-x))

square root(4-x)+((1x))/((2 square root 4-x))

I think I differentiated for it but how would I factor to find the critical numbers?

2. sin^2(x)+cos(x)

cos^2(x)-sin(x)

Here I have the same issue as my top problem

2. Re: Extrema on interval?

Hello, homeylova223!

What intervals are given?

$(1)\;\;f(x) \:=\:x\sqrt{4-x}$

Differentiate: . $f'(x) \;=\;1\cdot(4-x)^{\frac{1}{2}} + x\cdot\tfrac{1}{2}(4-x)^{-\frac{1}{2}}(-1)$

We have: . $\sqrt{4-x} - \frac{x}{2\sqrt{4-x}} \;=\;0$

Assume $x \ne 4$
Multiply by $2\sqrt{4-x}\!:\;\;2(4-x) - x \:=\:0$

. . Therefore: . $\boxed{x \:=\:\tfrac{8}{3}}$

$(2)\;\;f(x) \:=\:\sin^2\!x+ \cos x$

Differentiatte: . $2\sin x\cos x - \sin x$

We have: . $\sin x(2\cos x-1) \:=\:0$

And we have two equations to solve:

. . $\sin x \:=\:0 \quad\Rightarrow\quad \boxed{x \:=\:n\pi}$

. . $2\cos x-1\:=\:0 \quad\Rightarrow\quad \cos x \:=\:\tfrac{1}{2} \quad\Rightarrow\quad \boxed{x \:=\:\pm\tfrac{\pi}{3} + 2\pi n}$

3. Re: Extrema on interval?

Thanks your response was very helpful. Also the only interval given was for my second questions and it was between 0 and 2 pi I think.