Consider the function h(x) = 5x^3 - 3x^5

Question

(a) Find the intervals on which

f is increasing or decreasing.

(b) Find the local maximum and minimum values of

f.

Answer 1

Raj,

 

Where the h(x) increases the slope is positive and where it decreases the slope is negative.

And "the slope," once you get to calculus is also defined as the derivative (i.e. the function increases and decreases with + and - value of derivative respectively).

 

h′(x) = 5*3x²-3*5x⁴

=15x²(1-x²)

=15x²(1-x)(1+x)   difference of two squares

 

Draw a horizontal line and mark it at -1, 0, 1 and analyze the sign of the whole function in this four (x,-1;-1<x<0;0<x<1;x>1) regions as a result of the contribution from each factor.  

I will just do one of the four here and hope you would not have problem doing the other three. For x>1

x     -------- (+)

x     -------- (+) you have two xs and they both need to be taken into account

1-x  -------- (-)

1+x -------- (+),      (x)(x)(-)(x)=(-) decreasing

Can you do the rest?