Problem 10 - Essential Calculus

Question

Find the points of inflection and discuss the concavity of the graph of the function. \(f(x)=2 x^{4}-8 x+3\)

Step-by-Step Solution

Answer

The function does not have a point of inflection and it is concave upwards for all x in its domain.

Step 1: Find the derivative

The first step is to find the derivative of the function. The derivative of \(2x^4 - 8x + 3\) is \(f'(x) = 8 x^{3}-8\).

Step 2: Find the second derivative

Next, find the second derivative, by finding the derivative of \(f'(x)\). The second derivative of \(f(x)\) i.e. \(f''(x)\) is \(24x^2\).

Step 3: Find potential inflection points

The potential points of inflection are those values of \(x\) for which \(f''(x) = 0\). Solving \(f''(x) = 0\) yields \(x= 0\). There are no undefined points because \(f''(x)\) is a polynomial.

Step 4: Test the Points

Now test the nature of the function and determine its concavity by analyzing the sign of the second derivative on either side of 0. If \(x < 0\), \(f''(x) > 0\) which means that function is concave upward on the interval \(-\infty,0\) and if \(x > 0\), \(f''(x) > 0\), which means that the function is also concave upward on the interval \(0,+\infty\). Since the concavity doesn't change, \(x=0\) is not an inflection point.