Question

What is the maximum value that the graph of `f(x)= -x^2+8x+7`?

Answer 1

I got `23`.

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If you think about it, since `x^2` has a minimum, `-x^2 + bx + c` has a maximum (that doesn't require closed bounds).

We should know that:

• The slope is `0` when the slope changes sign.
• The slope changes sign when the graph changes direction.
• One way a graph changes direction is at a maximum (or minimum).
• Therefore, the derivative is `0` at a maximum (or minimum).

So, just take the derivative, set it equal to `0`, find the value of `x` (which corresponds to the maximum), and use `x` to find `f(x)`.

`f'(x) = -2x + 8` (power rule)

`0 = -2x + 8`

`2x = 8`

`color(blue)(x = 4)`

Therefore:

`f(4) = -(4)^2 + 8(4) + 7`

`= -16 + 32 + 7`

`=> color(blue)(y = 23)`

So your maximum value is `23`.

`y = -x^2 + 8x + 7`:

graph{-x^2 + 8x + 7 [-7.88, 12.12, 16.52, 26.52]}