Answer to Question #273986 in Calculus for Josa Kyme Palma

Question #273986


Find the points of inflection and discuss the concavity of the graph of the function

f(x)= (x+1)/√x


1
Expert's answer
2021-12-03T13:08:20-0500

Concavity and flection points are investigated using the second derivative and its sign."f(x)= \\frac{x+1}{ \\sqrt{x}}={x+1\\over x^{1\\over 2}}"

Apply quotient rule to get the first derivative :

Let "u=(x+1) \\implies u'=1" and "v=x^{1\\over 2} \\implies v'={1\\over 2}x^{-{1\\over 2}}"


"f'(x)={u'v-uv'\\over v^2}={1.x^{1\\over 2}-(x+1).({1\\over 2}x^{-{1\\over 2}})\\over (x^{1\\over 2})^2}={x-1\\over 2x^{3\\over 2}}"


"f''(x)=-{x-3\\over 4x^{5\\over 2}}=0 \\implies x=3"

If "x" is to the left of "3" (say at "x=2" ), "f''(x)\\gt 0" ,so the graph of "f(x)" is concave up to the left of 3.

If "x" is to the right of 3, "f''(x)\\lt0" ,so the graph of "f(x)" is concave down to the right of 3.

The concavity changes as we cross "x=3" if there is a point on a graph at "x=3" ( If 3 is in the domain of "f(x)" ) then that point is an inflection point

"f(3)={(3+1)\\over \\sqrt 3}=2.3"

So the point "(3,2.3)" is an infection point.


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