`f(x) = (x + 3)/(sqrt(x))` Find the points of inflection and discuss the concavity of the graph of the function.
Given the function `f(x)=(x+3)/sqrt(x)`
i.e it can be written as `f(x)=x/sqrt(x)+3/sqrt(x)` =`x^(1/2)+3x^(-1/2)`
Taking the first derivative we get,
Now again differentiating we get,
Now the points of inflection can be found out by equating f''(x) to zero. i.e.
Therefore x=9 is the inflection point.
Now choose auxilliary points x=8 to the left of inflection point and x=10 to the right of inflection point.
Therefore on (-infinity, 9) , the curve is concave upward .
Therefore on (9, infinity) , the curve is concave downward.
Find the second derivative.
Now we need to find the value(s) of x that make the second derivative zero:
Try x values on both sides, like 8 and 10
The concavity does change so `x=9 ` is a point of inflection.
Find the y-value, `(9,4) `
`(9,4) ` is a point of inflection