Question

Does the function `f(x)= -5x^2+x` have a minimum or maximum value?

Answer 1

`f(x)` has an absolute maximum of `0.05` at `x=0.1`

Explanation:

`f(x) =-5x^2+x`

`f(x)` is a parabola and will therefore have a single critical value. Since the coefficient of `x^2` is negative `f(x)` will have a single maximum. (NB: This answers the question as asked)

To determine the critical point I will use differential calculus (possibly outside the scope of this question)

`f'(x) = -10x+1`

`f_"max" (x)` will occur where `f'(x) =0`

`-10x+1=0 -> x=1/10= 0.1`

`f_"max" (x) = f(1/10) = -5xx1/100+1/10`

`=5/100 = 0.05`

The critical point can be seen on the graph of `f(x)` below.

graph{-5x^2+x [-0.6945, 0.638, -0.3875, 0.279]}