Let f:(0, π)→Rbe a twice differentiable function such that (lim)t→xt−xf(x)sint−f(x)sinx=sin2xfor all x∈(0, π). If f(6π)=−12π, then which of the following statement(s) is (are) TRUE?f(4π)=42π(b) f(x)<6x4−x2for all x∈(0, π)(c) There exists α∈(0, π)such that fprime(α)=0(d) f(2π)+f(2π)=0

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Let {f}:{({0}, )}{R}be a twice differentiable function such th

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Let $f : (0, π) \to R$ be a twice differentiable function such that $(lim)_{t \to x} \frac{f ( x ) sin t - f ( x ) sin x}{t - x} = sin^{2} x$ for all $x \in (0, π)$ . If $f (\frac{π}{6}) = - \frac{π}{12}$ , then which of the following statement(s) is (are) TRUE? $f (\frac{π}{4}) = \frac{π}{4 2}$ (b) $f (x) < \frac{x ^{4}}{6} - x^{2}$ for all $x \in (0, π)$ (c) There exists $α \in (0, π)$ such that $f^{p r i m e} (α) = 0$ (d) $f (\frac{π}{2}) + f (\frac{π}{2}) = 0$

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Amelia Discussed

Let

f : (0, π) \to R

be a twice differentiable function such that

(lim)_{t \to x} \frac{f ( x ) sin t - f ( x ) sin x}{t - x} = sin^{2} x

for all

x \in (0, π)

. If

f (\frac{π}{6}) = - \frac{π}{12}

, then which of the following statement(s) is (are) TRUE?

f (\frac{π}{4}) = \frac{π}{4 2}

(b)

f (x) < \frac{x ^{4}}{6} - x^{2}

for all

x \in (0, π)

α \in (0, π)

such that

f^{p r i m e} (α) = 0

(d)

f (\frac{π}{2}) + f (\frac{π}{2}) = 0

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Question Text	Let $f : (0, π) \to R$ be a twice differentiable function such that $(lim)_{t \to x} \frac{f ( x ) sin t - f ( x ) sin x}{t - x} = sin^{2} x$ for all $x \in (0, π)$ . If $f (\frac{π}{6}) = - \frac{π}{12}$ , then which of the following statement(s) is (are) TRUE? $f (\frac{π}{4}) = \frac{π}{4 2}$ (b) $f (x) < \frac{x ^{4}}{6} - x^{2}$ for all $x \in (0, π)$ (c) There exists $α \in (0, π)$ such that $f^{p r i m e} (α) = 0$ (d) $f (\frac{π}{2}) + f (\frac{π}{2}) = 0$
Answer Type	Video solution: 1
Upvotes	150