Respuesta :
The correct answer is option (a) f is continuous and differentiable at x = 5
A function is differentiable at a point if left hand limit = right hand limit at that point.
f(5) = -22 on solving
left hand limit (LHL) = [tex]lim_{x- > 5^{-} }[/tex] (f(x) - f(5)) / x - 5 = [tex]lim_{x- > 5^{-} }[/tex] (-x² + 3 + 22 ) / ( x - 5) = [tex]lim_{x- > 5^{-} }[/tex] (- ( x² + 25))/ (x- 5) = [tex]lim_{x- > 5^{-} }[/tex] (- (x-5)(x + 5)) / (x-5) = [tex]lim_{x- > 5^{-} }[/tex] (-(x+5) = - 10 (putting x = 5)
Right hand limit (RHL) = [tex]lim_{x- > 5 }[/tex] (f(x) - f(5) )/ ( x - 5) = [tex]lim_{x- > 5 }[/tex] (-10x + 28 + 22) / (x - 5) = [tex]lim_{x- > 5 }[/tex] ( - 10x + 50 )/ ( x - 5 ) = [tex]lim_{x- > 5 }[/tex] ( - 10 ( x - 5)) / (x-5 ) = -10
Thus on solving we see that LHL = RHL =-10 at the point 5
Therefore the function is differentiable at point 5
Also for the point say c = 5 , f(c) = -22 that is f(c) exists. The limit [tex]lim_{x- > 5 }[/tex] f(x) = - 22 and [tex]lim_{x- > 5^{-} }[/tex] f(x) = -22. As both the limits are equal so the limit exists. But we see that the limit of the function as x approaches c and the value of f(c) are equal. Thus the function is continuous.
Thus the correct answer to the question is option (a) that is f is continuous and differentiable at x = 5
What is a continuous function :
brainly.com/question/18102431
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