Question

Find the maximum value of the expression given below on the horizontal span of 0 to 6.    6x+7-x^2

Answer 1

f(x) = -1*x^2 + 6*x + 7
Differentiate with respect to x
f'(x) = d/dx(-1*x^2 + 6*x + 7)
= -2*x + 6

To find endpoints, we need to equate this to 0
So,
-2*x + 6 = 0
x = 3

Lets find double differentiation
f'(x) = -2*x + 6
Differentiate with respect to x
f''(x) = d/dx(-2*x + 6)
= -2

Lets find value of double differentiation at all endpoints
f''(x) = -2
f''(3) = -2
= -2
Since this is negative, x = 3 is maxima

Lets calculate the value of f(x) at x=0, 3 and 6

f(x) = -1*x^2 + 6*x + 7
f(0) = 0 + 0 + 7
= 7

f(3) = -(3)^2 + 6*3 + 7
= -9 + 18 + 7
= 16

f(6) = -(6)^2 + 6*6 + 7
= -36 + 36 + 7
= 7

So,
maximum value of 16