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(x)=2exaex+(2a+1)x3f\left( x \right) =2{ e }^{ x }-a{ e }^{ -x }+\left( 2a+1 \right) x-3 monotonically increasesxϵR\forall x\epsilon R ,then find the range of values of aa

(2) If f(x)=e2xaex+1f\left( x \right) ={ e }^{ 2x }-a{ e }^{ x }+1, prove that f(x)f\left( x \right) cannot be monotonically decreases for xϵR\forall x\epsilon R for any value of aa.

(3) The values of aa for which f(x)=(a+2)x3ax2=9ax1f\left( x \right) =\left( a+2 \right) { x }^{ 3 }-a{ x }^{ 2 }=9ax-1 monotonically decreasing.

(4) Let f(x)= {x2+x;1x<0λ;x=0log1/2(x+12);0<x<32}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 λ=0\lambda=0 and λ=1\lambda =1. For what values of =0=0 does f(x)f\left( x \right) has global maxima?

Note by Anandhu Raj
3 years, 10 months ago

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(1) f(x)=2exaex+(2a+1)x3f(x) = 2e^x - ae^{-x} + (2a + 1)x - 3
A function is said to be monotonically increasing if f(x)f'(x) > 0 for all real x .
f(x)=2ex+aex+(2a+1)>0\Rightarrow f'(x) = 2e^x + ae^x + (2a+1) > 0
a<2ex+1ex2\Rightarrow a < \dfrac{2e^x +1}{-e^x -2}
12<a<(From the graph)\Rightarrow -\dfrac{1}{2} < a < \infty \quad \quad \quad (\text{From the graph})

Similarly, you can try other parts.

Akhil Bansal - 3 years, 10 months ago

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