Determine the intervals on which the function f(x) is increasing and decreasing

Hi there

I am struggeling to compute this

I need to get the intervals where f(x) is increasing rounded to 4 decimal values as output

I realise it will be where f'(x)>0 and f'(x)<0

this is what i got sofar:

clf
f  = sinh(x.^2);
h  = sqrt(cosh(x)-1);
fplot(diff(f)-diff(h))
grid on
ylim([-5 5])
xlim([-5 5])

Answer 1

[Walter Roberson]

syms x
f  = sinh(x.^2);
g = diff(f);
sol = solve(g == 0, 'returnconditions', true)

sol = struct with fields:
             x: [3×1 sym]
    parameters: [1×1 sym]
    conditions: [3×1 sym]

sol.x

ans = 

sol.conditions

ans = 

by inspection: (-1)^(1/4) is complex-valued. sqrt(k+1/2) is complex valued for integer k <= -1, and complex times the complex root of unity is potentially real-valued. For integer k = 0 and upwards, sqrt(k+1/2) is real valued, and real times the complex root of unity is always complex-valued.

Therefore, we only need to consider k = -1 and less

assume(sol.parameters, 'integer')
curl = simplify(subs(diff(g), x, sol.x))

curl = 

The signs of those are the important parts: positive for local minima, negative for local maxima. For negative k, 2*k+1 is always negative, and -2*pi is always negative, so the sign comes down to the sign of (-1)^k which is negative for odd k (local maxima) and positive for even k (local minima)

sol.x tells you where the critical points are; curl tells you the maxima / minima. If you are at a local maxima, then everything to the next local minima (greater x, so decreasing k) is decreasing; if you are at a local minima, then everything until the next local maxima (greater x, so decreasing k) is increasing.

[Duncan Weakley]

Thank you so much!

This was a great explanation!

Extremely thankful for the help, I've been struggeling with this problem for 2 days.

Answer 2

[tarun Kumar v]

clf
f  = sinh(x.^2);
h  = sqrt(cosh(x)-1);
fplot(diff(f)-diff(h))
grid on
ylim([-5 5])
xlim([-5 5])