Question

Find the points of inflection and discuss the concavity of the graph of the function. f(x) = (x + 8) / ??x Step 1 Let f be a function whose second derivative exists on an open interval I. Then if f''(x) > 0 for all x in I, then the graph of f is concave up on I. And if f''(x) < 0 for all x in I, then the graph of f is concave down on I. If a tangent line exists at a point where concavity changes, this point is called a point of inflection. The given function is f(x) = (x + 8) / ??x. The domain of the function is x > 0. Step 2 To differentiate the given function with respect to x, perform the division in the original function and write with rational exponents. f(x) = (x + 8) / ??x = ??x + 8 / ??x = x^(1/2) + 8x^_ Take the derivative. f'(x) = 1/2x^_ - 4x^_

          Find the points of inflection and discuss the concavity of the graph of the function.
f(x) = (x + 8) / ??x

Step 1
Let f be a function whose second derivative exists on an open interval I. Then if f''(x) > 0 for all x in I, then the graph of f is concave up on I. And if f''(x) < 0 for all x in I, then the graph of f is concave down on I. If a tangent line exists at a point where concavity changes, this point is called a point of inflection.
The given function is f(x) = (x + 8) / ??x.
The domain of the function is x > 0.

Step 2
To differentiate the given function with respect to x, perform the division in the original function and write with rational exponents.
f(x) = (x + 8) / ??x = ??x + 8 / ??x = x^(1/2) + 8x^_
Take the derivative.
f'(x) = 1/2x^_ - 4x^_

Find the points of inflection and discuss the concavity of the graph of the function.
f(x) = (x + 8) / ??x

Step 1
Let f be a function whose second derivative exists on an open interval I. Then if f”(x) > 0 for all x in I, then the graph of f is concave up on I. And if f”(x) < 0 for all x in I, then the graph of f is concave down on I. If a tangent line exists at a point where concavity changes, this point is called a point of inflection.
The given function is f(x) = (x + 8) / ??x.
The domain of the function is x > 0.

Step 2
To differentiate the given function with respect to x, perform the division in the original function and write with rational exponents.
f(x) = (x + 8) / ??x = ??x + 8 / ??x = x^(1/2) + 8x^
Take the derivative.
f'(x) = 1/2x^ - 4x^

Added by Christopher W.

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