I've started studying the applications of derivatives and I'm not so fluent about the concepts. Could you please help me? Here are the some problems I'm struggling with,

(1) If \(f\left( x \right) =2{ e }^{ x }-a{ e }^{ -x }+\left( 2a+1 \right) x-3\) monotonically increases\(\forall x\epsilon R\) ,then find the range of values of \(a\)

(2) If \(f\left( x \right) ={ e }^{ 2x }-a{ e }^{ x }+1\), prove that \(f\left( x \right) \) cannot be monotonically decreases for \(\forall x\epsilon R\) for any value of \(a\).

(3) The values of \(a\) for which \(f\left( x \right) =\left( a+2 \right) { x }^{ 3 }-a{ x }^{ 2 }=9ax-1\) monotonically decreasing.

(4) Let \(f\left( x \right) =\ \begin{Bmatrix} { x }^{ 2 }+x\quad \quad ;-1\le x<0 \\ \lambda ;x=0 \\ \log _{ 1/2 }{ ( x+\frac { 1 }{ 2 } ) ;0<x<\frac { 3 }{ 2 } } \end{Bmatrix}\) . Discuss global maxima, minima for \(\lambda=0 \) and \(\lambda =1\). For what values of \(=0\) does \(f\left( x \right) \) has global maxima?

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TopNewest(1) \(f(x) = 2e^x - ae^{-x} + (2a + 1)x - 3 \)

A function is said to be monotonically increasing if \(f'(x)\) > 0 for all real x .

\(\Rightarrow f'(x) = 2e^x + ae^x + (2a+1) > 0 \)

\(\Rightarrow a < \dfrac{2e^x +1}{-e^x -2}\)

\(\Rightarrow -\dfrac{1}{2} < a < \infty \quad \quad \quad (\text{From the graph})\)

Similarly, you can try other parts. – Akhil Bansal · 1 year, 8 months ago

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