Question

How do you determine whether the function `f(x)= -6 sqrt (x)` is concave up or concave down and its intervals?

Answer 1

Use calculus (the sign of the second derivative) or algebra/precalculus graphing techniques.

Explanation:

Calculus

In general, to investigate concavity of the graph of function `f`, we investigate the sign of the second derivative.

`f(x) = -6x^(1/2)`

Note first that the doamin of `f` is `[0,oo)`

`f'(x) = -3x^(-1/2)`

`f''(x) = 3/2 x^(-3/2) = 3/(2sqrtx^3)`

`f''(x)` is positive for all real `x`, so it is positive for all `x` in the domain of `f`.

The graph of `f` is concave up on `(0,oo)`.

(Intervals of concavity are generally given as open intervals.)

Algebra/Precalculus

The graph of the square root function looks like this:

graph{y = sqrtx [-10, 10, -5, 5]}

That graph is concave down.

Multiplying by `-6` reflects the graph across the `x` asix and stretches it vertically by a factor of `6`:

graph{y = -6sqrtx [-6.41, 25.63, -13.2, 2.82]}

The graph is concave up.