A function is defined in 0- - by fx - 1 - x2 for 0 - x - 1 ln x for 1 - x - 2 ln 2 - 1 - 0-5x for 2 - x - -Which one of the following is correct in respect of the derivative of the function- i-e- f-x-

The correct option is B f'(x) = 2x #160; for #160; 0 lt; x ≤ 1 f(x) is continuous at x=1 and at x=2 .Differentiating w.r.t ...

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Standard XII

Mathematics

Continuity in an Interval

A function is...

Question

A function is defined in $(0, \infty)$ by
$f (x) = ⎧ ⎪ ⎨ ⎪ ⎩ \begin{matrix} 1 - x^{2} & f o r & 0 < x \leq 1 ln x & f o r & 1 < x \leq 2 ln 2 - 1 + 0.5 x & f o r & 2 < x < \infty \end{matrix}$
Which one of the following is correct in respect of the derivative of the function, i.e., $f^{'} (x)$ ?

f^{'} (x) = 2 x

for

0 < x \leq 1

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f^{'} (x) = - 2 x

for

0 < x \leq 1

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f^{'} (x) = - 2 x

for

0 < x < 1

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f^{'} (x) = 0

for

0 < x < \infty

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Solution

The correct option is B $f^{'} (x) = - 2 x$ for $0 < x \leq 1$
$f (x)$ is continuous at $x = 1$ and at $x = 2$ .
Differentiating w.r.t $x$
$f^{'} (x) = ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ \begin{matrix} - 2 x & for & 0 < x \leq 1 \frac{1}{x} & for & 1 < x \leq 2 0.5 & for & 2 < x < \infty \end{matrix} ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭$

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A function is defined in (0,∞) byf(x)=⎧⎪⎨⎪⎩1−x2for0<x≤1ln xfor1<x≤2ln 2−1+0.5xfor2<x<∞Which one of the following is correct in respect of the derivative of the function, i.e., f′(x)?

The correct option is B f′(x)=−2x for 0<x≤1f(x) is continuous at x=1 and at x=2.Differentiating w.r.t xf′(x)=⎧⎪ ⎪ ⎪⎨⎪ ⎪ ⎪⎩−2xfor0<x≤11xfor1<x≤20.5for2<x<∞⎫⎪ ⎪ ⎪⎬⎪ ⎪ ⎪⎭