1. Tardigrade
  2. Question
  3. Mathematics
  4. Give the correct order of initials T or F for following statements. Use T if statement is true and F if it is false. Statement-1: If f: R arrow R and c ∈ R is such that f is increasing in (c-δ, c) and f is decreasing in (c, c+δ) then f has a local maximum at c. Where δ is a sufficiently small positive quantity. Statement-2: Let f :( a , b ) arrow R , c ∈( a , b ). Then f can not have both a local maximum and a point of inflection at x = c. Statement-3: The function f ( x )= x 2| x | is twice differentiable at x =0. Statement-4: Let f :[ c -1, c +1] arrow[ a , b ] be bijective map such that f is differentiable at c then f -1 is also differentiable at f ( c ).

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Q. Give the correct order of initials $T$ or $F$ for following statements. Use $T$ if statement is true and $F$ if it is false.
Statement-1: If $f: R \rightarrow R$ and $c \in R$ is such that $f$ is increasing in $(c-\delta, c)$ and $f$ is decreasing in $(c, c+\delta)$ then $f$ has a local maximum at $c$. Where $\delta$ is a sufficiently small positive quantity.
Statement-2 : Let $f :( a , b ) \rightarrow R , c \in( a , b )$. Then $f$ can not have both a local maximum and a point of
inflection at $x = c$.
Statement-3: The function $f ( x )= x ^2| x |$ is twice differentiable at $x =0$.
Statement-4 : Let $f :[ c -1, c +1] \rightarrow[ a , b ]$ be bijective map such that $f$ is differentiable at $c$ then $f ^{-1}$ is also differentiable at $f ( c )$.

Application of Derivatives

Solution:

S-1 : f must be continuous $\Rightarrow$ false $S -2 x = c$ is local max.but not inflection point $\Rightarrow$ True

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