maximum point

Find the coordinates of the maximum point on the curve y = -2xsquared + 6x + 3. Explain how you know it is a maximum?

f'(x) = -4 x + 6
so maximum point is when f'(x) = 0 so
-4x + 6 = 0
-4 x = -6
x = 1.5

Sustituing we get

y = 7.5

so (1.5, 7.5) is this correct?

How can I garanty it being the maximum point though?

Quote:

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Originally Posted by Natasha1

so (1.5, 7.5) is this correct?

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Yes. (Yes)

Please explain how you know it is a maximum?


Moderator edit: Added "Please" and deleted excessive ?'s.

Quote:

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Originally Posted by Natasha1

Explain how you know it is a maximum????

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Say the function has a stationary point at . Then the second derivative test states that is a
maximum if , and a minimum if , and a possible point of inflexion if .
You have , thus . If , we still have .

so the second derivative of f'(x) = -4 x + 6 would be

f''(x) = -4 which is < 0 and hence it is by definition the maximum point

Quote:

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Originally Posted by Natasha1

so the second derivative of f'(x) = -4 x + 6 would be

f''(x) = -4 which is < 0 and hence it is by definition the maximum point

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That's right.