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Chapter 3: Q. 1 (page 313)

Fill in the blanks to complete each of the following theorem statements:

  • If x=cis a local extremum of f, then localid="1648527084118" f'(c)is either _______or_______.

Short Answer

Expert verified

Ans: part (a). f+'(c)=limx鈫�c+f(x)-f(c)x-c鈮�0

part (b).f-'(c)=limx鈫�c-f(x)-f(c)x-c鈮�0

Step by step solution

01

Step 1. Given Information:

For a functionf(x)the local extremum isx=c

02

Step 2. Explanation:

  • Suppose x=cis the location of a local maximum of f.
  • If f'(c)does not exist, then x=cis a critical point .
  • And if f'(c)exists, then it must be equal to 0.
Sincex=cisthelocationofalocalmaximum未>0鈭�x鈭�(c鈭�未,c+未)wheref(c)鈮�f(x)thus,f(x)鈭�f(c)鈮�0Inthecasewherex鈭�(c,c+未)......[sox>c]鈭�x鈭�cispositivecasef(x)鈭�f(c)x鈭�c鈮�0f'+(c)=limx鈫�c+f(x)鈭�f(c)x鈭�c鈮�0similarlyx鈭�(c鈭�未,c)......[sox<c]andf(x)鈭�f(c)鈮�0,f'-(c)=limx鈫�c-f(x)鈭�f(c)x鈭�c鈮�0

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Most popular questions from this chapter

Find the possibility graph of its derivative f'.

Restate Theorem 3.3 so that its conclusion has to do with

tangent lines.

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role="math" localid="1648370582124" f(x)=sinx.cosx
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