Maximum value

Question

what is the maximu value for the function f(x)=-x^2+2x+2?

Answer 1

You can also determine the maximum without resorting to Calculus.

 

y = -x²+2x+2 is a quadratic function.  The graph of a quadratic function is a parabola.  

 

In general, for y = Ax²+Bx+C, if A < 0, then the graph is a parabola opening downward with maximum value when x = -B/(2A).  

 

For the given quadratic function, the maximum occurs when

    x = -2/(2(-1)) = 1

 

Maximum value = f(1) = -(1)²+2(1)+2 = 3

 

 

Answer 2

The maximum value would exist for that value of x where the derivative of the function is equal to 0.

 

Let f(x) = y

 

Then y = -x² + 2x + 2

 

Calculate the derivative dy/dx

 

dy/dx = -2x + 2

 

Set the derivative equal to 0 and solve for x

 

-2x + 2 = 0

 

x = 1

 

This value of x yields a maximum value for the function f(x) which is 3.

 

You can test it by plugging in values around 1 (such as  -1, 0, , 2, etc) and seeing that they yield function values  less than the maximum.