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Basics: Applications of derivatives

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?

Note by Anandhu Raj
1 year, 2 months ago

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(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. · 1 year, 2 months ago

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