Find the maximum value

Question

y=-3x^2-9x+2

Answer 1

Since the coefficient of the x^2 term is negative, this is an equation of a downward facing parabola, whose maximum value is found at its vertex.

 

Method 1 (Algebra)

Since, y=-3x^2-9x+2              eq1

Coefficients of this equations would be: a = -3, b=-9, c=2

 

If the vertex is (h,k), then h = -b/2a = 9/-6 = -3/2

Since point (h,k) lies on the parabola, it should satisfy eq1

k=-3h^2-9h+2

  = -3(9/4) -9(-3/2)+2

  = -27/4 + 27/2 + 2

  = 27/4 + 2

  = 35/4

Therefore, the vertex of the parabola = (-3/2, 35/4)

and maximum value = 35/4

 

Method2 (Calculus)

An easier way of finding the max value is to use the calculus principles.

At the max point the slope of the line would be zero, meaning

dy/dx = 0

d/dx(-3x^2-9x+2) = 0

-6x-9 = 0

x = -3/2

So, max value is found at x=-3/2

To calculate it, we substitute x=-3/2 in eq1

y_max = -3(9/4) -9(-3/2) + 2

         = 35/4