Question

I need help with this word problem please

Answer 1

Answer:

Step-by-step explanation:

Given function is,

f(x) = $x\sqrt{x}$

We can rewrite this function as,

f(x) = $x^{\frac{3}{2}}$

a). Domain of the function is [0, ∞)

Derivative of the function in this domain,

f'(x) = $\frac{3}{2}x^{(\frac{3}{2}-1)}$

     = $\frac{3}{2}x^{\frac{1}{2}}$

Since, f'(x) > 0 in the interval [0, ∞)

Function is increasing in the interval [0, ∞)

b). For local maxima and local minima,

  Second derivative of the function,

  f"(x) = $\frac{3}{2}(\frac{1}{2})x^{\frac{1}{2}-1}$

         = $\frac{3}{4}x^{-\frac{1}{2}}$  > 0

   So the function should have a local minima.

   But the function is continuous in the interval x ≥ 0,

   There is no local minima (only absolute minima).

c). Since, second derivative is undefined at x = 0,

   There are no inflection points.

d). Since, f"(x) > 0 in the interval [0, ∞)

   Concavity of the curve is UPWARDS.