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Romashka-Z-Leto [24]
3 years ago
12

Determine whether Rolle's Theorem can be applied to f on the closed interval

Mathematics
2 answers:
notsponge [240]3 years ago
8 0

Rolle's theorem is applicable if f(a)=f(b) and $f$ is differentiable in $(a,b)$

since it's polynomial function, it's always continuous and differentiable..

and you can easily check that $f(0)=f(-3)=0$

so it is applicable.

now, $f'(x)=-2x+3=0 \implies x=\frac32$

there is only once value (as you can imagine, the graph will be downward parabola)

Ierofanga [76]3 years ago
6 0

Answer:

Yes, Rolle's theorem can be applied

There is only one value of c such that f'(c) = 0, and this is c = 1.5 (or 3/2 in fraction form)

Step-by-step explanation:

Yes, Rolle's theorem can be applied on this function because the function is continuous in the closed interval (it is a polynomial function) and differentiable  in the open interval, and f(a) = f(b) given that:

f(0)=-0^2+3\,(0)=0\\f(3)=-3^2+3\,(3)=-9+9=0

Then there must be a c in the open interval for which f'(c) =0

In order to find "c", we derive the function and evaluate it at "c", making the derivative equal zero, to solve for c:

f(x)=-x^2+3\,x\\f'(x)=-2\,x+3\\f'(c)=-2\,c+3\\0=-2\,c+3\\2\,c=3\\c=\frac{3}{2} =1.5

There is a unique answer for c, and that is c = 1.5

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