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Question

Let $f left parenthesis x right parenthesis equals left curly bracket to the power of x squared vertical line blank cos blank pi over chi vertical line end exponent right curly bracket 0 comma$ $x equals 0 x not equal to 0$ $chi element of R$ Then $f$ is

Differentiable both at x=0 and at x=2
Differentiable at $x = 0$ but not differentiable at x=2
Not differentiable at x=0 but differentiable at x=2
Differentiable neither at x=0 nor at x=2

The correct answer is: Differentiable at $x equals 0$ but not differentiable at x=2

$f to the power of ´ end exponent left parenthesis 0 right parenthesis equals h rightwards arrow 0 vertical line i m fraction numerator f left parenthesis 0 plus h right parenthesis minus f left parenthesis 0 right parenthesis over denominator h end fraction$
$equals h rightwards arrow 0 vertical line i m fraction numerator h to the power of 2 end exponent vertical line blank c o s blank fraction numerator pi over denominator h end fraction vertical line minus 0 over denominator h end fraction$
$equals h rightwards arrow 0 vertical line i m h blank c o s blank left parenthesis fraction numerator pi over denominator h end fraction right parenthesis equals 0$
so, $f left parenthesis x right parenthesis$ is differentiable at $x equals 0$
$f to the power of ´ end exponent left parenthesis 2 to the power of plus end exponent right parenthesis equals h rightwards arrow 0 vertical line i m fraction numerator f left parenthesis 2 plus h right parenthesis minus f left parenthesis 2 right parenthesis over denominator h end fraction$
$equals h rightwards arrow 0 vertical line i m fraction numerator left parenthesis 2 plus h right parenthesis to the power of 2 end exponent vertical line blank c o s blank fraction numerator pi over denominator 2 plus h end fraction vertical line minus 0 over denominator h end fraction$
$equals h rightwards arrow 0 vertical line i m fraction numerator left parenthesis 2 plus h right parenthesis to the power of 2 end exponent blank c o s blank left parenthesis fraction numerator pi over denominator 2 plus h end fraction right parenthesis over denominator h end fraction$
$f to the power of ´ end exponent left parenthesis 2 to the power of plus end exponent right parenthesis equals h rightwards arrow 0 vertical line i m fraction numerator left parenthesis 2 plus h right parenthesis to the power of 2 end exponent over denominator h end fraction blank s i n blank left parenthesis fraction numerator pi over denominator 2 end fraction minus fraction numerator pi over denominator 2 plus h end fraction right parenthesis$
$equals stack l i m with h rightwards arrow 0 below fraction numerator left parenthesis 2 plus h right parenthesis to the power of 2 end exponent over denominator h end fraction blank s i n blank left square bracket fraction numerator pi times h over denominator 2 left parenthesis 2 plus h right parenthesis end fraction right square bracket$
null
Again, $f to the power of ´ end exponent left parenthesis 2 to the power of minus end exponent right parenthesis equals stack lim with h rightwards arrow 0 below invisible function application fraction numerator f left parenthesis 2 minus h right parenthesis minus f left parenthesis 2 right parenthesis over denominator negative h end fraction$
$equals stack l i m with h rightwards arrow 0 below fraction numerator left parenthesis 2 minus h right parenthesis to the power of 2 end exponent 1 blank c o s blank left parenthesis fraction numerator pi over denominator 2 minus h end fraction right parenthesis divided by over denominator negative h end fraction$
$equals stack l i m with h rightwards arrow 0 below fraction numerator negative left parenthesis 2 minus h right parenthesis to the power of 2 end exponent blank c o s blank left parenthesis fraction numerator pi over denominator 2 minus h end fraction right parenthesis over denominator negative h end fraction$
$equals stack l i m with h rightwards arrow 0 below fraction numerator left parenthesis 2 minus h right parenthesis to the power of 2 end exponent blank s i n blank left square bracket fraction numerator pi over denominator 2 end fraction minus fraction numerator pi over denominator 2 minus h end fraction right square bracket over denominator h end fraction$
$equals stack l i m with h rightwards arrow 0 below fraction numerator left parenthesis 2 minus h right parenthesis to the power of 2 end exponent over denominator h end fraction times blank s i n blank left square bracket fraction numerator negative pi h over denominator 2 left parenthesis 2 minus h right parenthesis end fraction right square bracket$
$equals negative stack l i m with h rightwards arrow 0 below fraction numerator left parenthesis 2 minus h right parenthesis to the power of 2 end exponent over denominator pi h end fraction blank s i n blank fraction numerator pi h over denominator 2 left parenthesis 2 minus h right parenthesis end fraction cross times fraction numerator pi over denominator 2 left parenthesis 2 minus h right parenthesis end fraction equals negative pi$
$Error converting from MathML to accessible text.$
Hence, the correct answer is option (B).

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