Explanation

f(x) = e^x - ax makes a U-shape because e^x is always curving up. 

U-shape has only 1 lowest point at x = ln a. 

So the minimum value = f(ln a) = a - a ln a

Last verified on 04-06-2026

Explanation

System:

x1 + 2x2 + x3 = 9

2x1 + x2 + 3x3 = 7

Matrix form AX = B:

[1 2 1] [x1] [9]

[2 1 3] [x2] = [7]

          [x3]

Augmented matrix [A|B]:

[1 2 1 | 9]

[2 1 3 | 7]

Last verified on 04-06-2026

Explanation

x = A cos(mt - a) is a wave/oscillation.

If you differentiate it twice, you get the same x back but with a minus sign and m² in front.

So the equation becomes: d²x/dt² = -m²x

Clear rule: Cos or Sin → 2nd derivative = -ω² × function

Last verified on 04-06-2026

Explanation

Explanation

Last verified on 04-06-2026