determine the point of inflection, if it exists, on the graph of f(x)=5x^3+4x^(5/2)

Answer 1

f(x) = 5x³ + 4x^5/2

Note that x^5/2 = (√x)⁵. So, f is defined only when x≥0.

f'(x) = 15x² + 10x^3/2

f"(x) = 30x + 15x^1/2 = 15x^1/2(2√x + 1) > 0 for all x > 0

So, f is concave up for x > 0. Therefore, f has no inflection points.

Answer 2

To find inflection point, set 2nd derivative = 0. f'' = 30x + 15x^1/2

The second derivative is 0 when x = 0

I think this is not a point of inflection for this function as this is the limit of the domain.(x>=0)