1. ## 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?

2. Originally Posted by Natasha1
so (1.5, 7.5) is this correct?
Yes.

3. Please explain how you know it is a maximum?

4. Originally Posted by Natasha1
Explain how you know it is a maximum????
Say the function $\displaystyle f(x)$ has a stationary point at $\displaystyle \xi$. Then the second derivative test states that $\displaystyle \xi$ is a
maximum if $\displaystyle f''(\xi)<0$, and a minimum if $\displaystyle f''(\xi)>0$, and a possible point of inflexion if $\displaystyle f''(\xi) = 0$.
You have $\displaystyle f'(x) = -4 x + 6$, thus $\displaystyle f''(x) = -4$. If $\displaystyle x = 7.5$, we still have $\displaystyle f''(x) = -4 <0$.

5. 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

6. 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
That's right.