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lisov135 [29]
3 years ago
6

Find the critical numbers of the function. (Enter your answers as a comma-separated list. Use n to denote any arbitrary integer

values. If an answer does not exist, enter DNE.) f(θ) = 10 cos(θ) + 5 sin2(θ)
Mathematics
1 answer:
lyudmila [28]3 years ago
3 0

If

f(\theta)=10\cos\theta+5\sin^2\theta

then the derivative is

f'(\theta)=-10\sin\theta+10\sin\theta\cos\theta

Critical points occur where f'(\theta)=0. This happens for

-10\sin\theta+10\sin\theta\cos\theta=0

-10\sin\theta(1-\cos\theta)=0

\implies-10\sin\theta=0\text{ or }1-\cos\theta=0

In the first case, we find

-10\sin\theta=0\implies\sin\theta=0\implies\theta=n\pi

In the second,

1-\cos\theta=0\implies\cos\theta=1\implies\theta=2n\pi

So all the critical points occur at multiples of \pi, or n\pi. (This includes all the even multiples of \pi.)

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