80216 Let \(f(x)\) be differentiable in \([2,5]\) and \(f^{\prime}(x) \geq\) \(\mathbf{4} / 3\) for all \(\mathrm{x} \in[2,5]\). If \(\boldsymbol{f}(\mathbf{2})=\mathbf{2}\), then

80217 Suppose \(\mathrm{f}: \mathrm{R} \rightarrow(0, \infty)\) be a differentiable function such that \(5 \mathrm{f}(\mathrm{x}+\mathbf{y})=\mathbf{f}(\mathbf{x}) . \mathbf{f}(\mathbf{y}), \forall \mathrm{x}, \mathrm{y}\) \(\in R\). If \(f(3)=320\), then \(\sum_{n=0}^{5} f(n)\) is equal to :

80218 Let \(f, g: R \rightarrow\) be two real valued functions defined as \(f(x)=\left\{\begin{aligned}-|x+3|, x\lt 0 \\ e^{x}, x>0\end{aligned}\right.\) and \(g(x)=\left\{\begin{array}{ll}x^{2}+k_{1} x, x\lt 0 \\ 4 x+k_{2}, x \geq 0\end{array}\right.\), where \(k_{1}\) and \(k_{2}\) are real constants. If (gof) is differentiable at \(x=0\), then (gof) (-4) \(+(\) gof \()(4)\) is equal to :

80219 The set of all points where the function \(f(x)=\) \(\mathbf{2 x}|\mathbf{x}|\) is differentiable is

80216 Let \(f(x)\) be differentiable in \([2,5]\) and \(f^{\prime}(x) \geq\) \(\mathbf{4} / 3\) for all \(\mathrm{x} \in[2,5]\). If \(\boldsymbol{f}(\mathbf{2})=\mathbf{2}\), then

80217 Suppose \(\mathrm{f}: \mathrm{R} \rightarrow(0, \infty)\) be a differentiable function such that \(5 \mathrm{f}(\mathrm{x}+\mathbf{y})=\mathbf{f}(\mathbf{x}) . \mathbf{f}(\mathbf{y}), \forall \mathrm{x}, \mathrm{y}\) \(\in R\). If \(f(3)=320\), then \(\sum_{n=0}^{5} f(n)\) is equal to :

80218 Let \(f, g: R \rightarrow\) be two real valued functions defined as \(f(x)=\left\{\begin{aligned}-|x+3|, x\lt 0 \\ e^{x}, x>0\end{aligned}\right.\) and \(g(x)=\left\{\begin{array}{ll}x^{2}+k_{1} x, x\lt 0 \\ 4 x+k_{2}, x \geq 0\end{array}\right.\), where \(k_{1}\) and \(k_{2}\) are real constants. If (gof) is differentiable at \(x=0\), then (gof) (-4) \(+(\) gof \()(4)\) is equal to :

80219 The set of all points where the function \(f(x)=\) \(\mathbf{2 x}|\mathbf{x}|\) is differentiable is

80216 Let \(f(x)\) be differentiable in \([2,5]\) and \(f^{\prime}(x) \geq\) \(\mathbf{4} / 3\) for all \(\mathrm{x} \in[2,5]\). If \(\boldsymbol{f}(\mathbf{2})=\mathbf{2}\), then

80217 Suppose \(\mathrm{f}: \mathrm{R} \rightarrow(0, \infty)\) be a differentiable function such that \(5 \mathrm{f}(\mathrm{x}+\mathbf{y})=\mathbf{f}(\mathbf{x}) . \mathbf{f}(\mathbf{y}), \forall \mathrm{x}, \mathrm{y}\) \(\in R\). If \(f(3)=320\), then \(\sum_{n=0}^{5} f(n)\) is equal to :

80218 Let \(f, g: R \rightarrow\) be two real valued functions defined as \(f(x)=\left\{\begin{aligned}-|x+3|, x\lt 0 \\ e^{x}, x>0\end{aligned}\right.\) and \(g(x)=\left\{\begin{array}{ll}x^{2}+k_{1} x, x\lt 0 \\ 4 x+k_{2}, x \geq 0\end{array}\right.\), where \(k_{1}\) and \(k_{2}\) are real constants. If (gof) is differentiable at \(x=0\), then (gof) (-4) \(+(\) gof \()(4)\) is equal to :

80219 The set of all points where the function \(f(x)=\) \(\mathbf{2 x}|\mathbf{x}|\) is differentiable is

80216 Let \(f(x)\) be differentiable in \([2,5]\) and \(f^{\prime}(x) \geq\) \(\mathbf{4} / 3\) for all \(\mathrm{x} \in[2,5]\). If \(\boldsymbol{f}(\mathbf{2})=\mathbf{2}\), then

80217 Suppose \(\mathrm{f}: \mathrm{R} \rightarrow(0, \infty)\) be a differentiable function such that \(5 \mathrm{f}(\mathrm{x}+\mathbf{y})=\mathbf{f}(\mathbf{x}) . \mathbf{f}(\mathbf{y}), \forall \mathrm{x}, \mathrm{y}\) \(\in R\). If \(f(3)=320\), then \(\sum_{n=0}^{5} f(n)\) is equal to :

80218 Let \(f, g: R \rightarrow\) be two real valued functions defined as \(f(x)=\left\{\begin{aligned}-|x+3|, x\lt 0 \\ e^{x}, x>0\end{aligned}\right.\) and \(g(x)=\left\{\begin{array}{ll}x^{2}+k_{1} x, x\lt 0 \\ 4 x+k_{2}, x \geq 0\end{array}\right.\), where \(k_{1}\) and \(k_{2}\) are real constants. If (gof) is differentiable at \(x=0\), then (gof) (-4) \(+(\) gof \()(4)\) is equal to :

80219 The set of all points where the function \(f(x)=\) \(\mathbf{2 x}|\mathbf{x}|\) is differentiable is