What is the minimum slope of y=x^3-9*x^2+15*x? matlab code

What is the minimum slope of y=x^3-9*x^2+15*x?

[madhan ravi]

what have you tried so far?

[Busra Tabak]

syms x
f=x^3-9*x^2+15*x;
fprime=diff(f,x);
fdprime=diff(fprime,x);
xstar=double(solve(fprime==0,x));
xstar=unique(xstar);
for i=1:numel(xstar)
    if subs(fdprime,x,xstar(i))>0
        disp('min')
    double(subs(f,x,xstar(i)))
    end
end
 

I wrote this code and got this solution. Is it correct?

min
 
ans =
 
   -25

Answer 1

[Image Analyst]

Hint:

slope = 3 * x .^ 2 - 18 * x + 15; % Derivative.
minSlope = min(slope)
minAbsSlope = min(abs(slope))

Plot it, using linspace() for x, and see what you see. Of course since it's a parabola, the min absolute value of the slope will be zero and the min slope will depend on how far negative you want to evaluate x. For x that is more negative, the slope will be steeper.

[Busra Tabak]

I do not understand and I can not write the code. Did you find -25 as answer?

[Image Analyst]

No, of course not. Did you just plot the function and see it? Nowhere does it turn downwards and have a negative slope:

x = linspace(-50, 50, 100000);
y = x.^3-9*x.^2+15*x;
subplot(2, 1, 1);
plot(x, y, 'b-')
grid on;
fontSize = 20;
title('y = x .^ 3 - 9 * x .^ 2 + 15 * x', 'FontSize', fontSize, 'Interpreter', 'none');
xlabel('x', 'FontSize', fontSize, 'Interpreter', 'none');
ylabel('y', 'FontSize', fontSize, 'Interpreter', 'none');
slope = 3 * x .^ 2 - 18 * x + 15; % Derivative.
minSlope = min(slope)
minAbsSlope = min(abs(slope))
subplot(2, 1, 2);
plot(x, slope, 'b-')
xlabel('x', 'FontSize', fontSize, 'Interpreter', 'none');
ylabel('Slope of y', 'FontSize', fontSize, 'Interpreter', 'none');
title('slope = 3 * x .^ 2 - 18 * x + 15', 'FontSize', fontSize, 'Interpreter', 'none');
grid on;

It's a parabola so what do you think the min absolute slope would be? And don't you see and understand that the min slope (most negative) will depend on how far out into the negative x territory you want to go?