Given the function g(x) = 8x2 - 48x2 + 72x,the first derivative, g\'(x). g\'(x)

Question

Given the function g(x) = 8x2 - 48x2 + 72x,the first derivative, g'(x). g'(x) = Preview Notice that g'(x) = 0 when x = 1, that is, g'(1) = 0 . Now, we want to know whether there is a local minimum or local maximum at x = 1, so we will use the second derivative test. Find the second derivative, g"(x)= Preview Evaluate g''(1) = Based on the sign of this number, does this mean the graph of g(x) is concave up or concave down at x = 1 ? [Answer either up or down -- watch your spelling!!] At x = 1 the graph of g(x) is concave Based on the concavity of g(x) at x = 1, does this mean that there is a local minimum or local maximum at x = 1? [Answer either minimum or maximum - watch your spelling!!] At x = 1 there is a local

Explanation / Answer

g'(x) = 24x^2 - 96x + 72 g''(x) = 48x - 96 g''(1) = -48 at x = 1 we get g''(1) < 0 so we have concave downwards at x = 1 we have local maxima as g''(x) < 0 at x = 1

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