Explanation

For a parabola y = ax² + bx + c, the x-coordinate of vertex = -b/(2a)

Here y = 2x² - 4x + 3  

a = 2, b = -4  

x_vertex = -(-4) / (2×2) = 4/4 = 1

Last verified on 07-06-2026

Explanation

x cot y dy = dx  

cot y dy = dx/x  

∫ cot y dy = ∫ dx/x  

ln|sin y| = ln|x| + C  

siny = Cx

Last verified on 05-06-2026

Explanation

Let width = W, length = L. Fence for 3 sides: L + 2W = x → L = x - 2W  

Area A = L×W = (x - 2W)W = xW - 2W²  

For max, dA/dW = x - 4W = 0 → W = x/4  

Then L = x - 2(x/4) = x/2  

Max area = L×W = (x/2)(x/4) = x²/8

Last verified on 05-06-2026

Explanation

Inflection point: y'' = 0  

y' = 3x² - 6x + 4  

y'' = 6x - 6 = 0 → x = 1  

At x = 1: y = 1 - 3 + 4 = 2  

Slope m = y' at x=1 = 3 - 6 + 4 = 1  

Tangent: y - 2 = 1(x - 1) → y = x + 1

Last verified on 05-06-2026

Explanation

Laplace of e^(at) = ∫ e^(-st) e^(at) dt from 0 to ∞  

= ∫ e^(-(s-a)t) dt = 1/(s - a), when s > a

Last verified on 05-06-2026