Question

For what value of k, the function f (x) = kx^3+ 5 is decreasing?​

Answer 1

Answer:

yep

Step-by-step explanation:

Given : f(x)=sinx−cosx−Kx+5

f(x) decrease for all x if f  

′

(x)<0  

f  

′

(x)=cosx+sinx−K<0

∴k>cosx+sinx

We know that, cosx+sinx=  

2

​  

(  

2

​  

 

1

​  

cosx+  

2

​  

 

1

​  

sinx)=  

2

​  

(sin  

4

π

​  

cosx+cos  

4

π

​  

sinx)=  

2

​  

(sin(  

4

π

​  

+x))

and,

−1≤sin(  

4

π

​  

+x)≤1⇒−  

2

​  

≤  

2

​  

⋅sin(  

4

π

​  

+x)≤  

2

​  

 

∴−  

2

​  

≤cosx+sinx≤  

2

​  

 

max(cosx+sinx)=  

2

​  

 

∴K>  

2

​

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