Suppose f″ is continuous on (−∞, ∞). If f′(2)=0 and f″(2)=−5, what can you say about f?

Question

Suppose f″ is continuous on (−∞, ∞). If f′(2)=0 and f″(2)=−5, what can you say about f?

Uersfeldte · Accepted Answer

Step 1Second Derivative Test: This test is applicable when f″ is contionuous near c1) If f′(c)=0 and f″(c)&amp;gt;0, then there is a local minimum at c2) If f′(c)=0 and f″(c)&amp;lt;0, then there is a local maximum at cNote that:1) It is given that f″ is continuous everywhere, so the second derivative test is applicable at x=22) It is given that f′(2)=0 and f″(2)&amp;lt;0Therefore, we can conclude that there is a local maximim at x=2

Annie Midgett · Accepted Answer

Step 1Because of the definition of a Critical Point, we know that in f(x), there is a critical point at x=2 becausef′(2)=0Furthermore, by the Second Derivative Test, we know that f(x) has a local maximum at x=2 because f″(2)&amp;lt;0f(2) is a critical point, and is a local maximum.

Suppose f'' is continuous on (-\infty,\ \infty). If f'(2)=0 and

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New Questions in Calculus 1