Maths Applications of Derivatives

Let α ∈ R and A be a matrix of order 3 × 3 such that det(A) = −4 and A + I = [1 a 1; 2 1 0; a 1 2], where I is the identity matrix of order 3 × 3.
If det((a + 1)adj((a – 1)A)) is 2?3?, m, n ∈ {0,1,2, ., 20}, then m + n is equal to:

The equation of the normal to the curve y = (1+x)²? + cos²(sin?¹x) at x=0 is

If p(x) be a polynomial of degree three that has a local maximum value 8 at x=1 and a local minimum value 4 at x=2; then p(0) is equal to

If a and b are unit vectors, then the greatest value of √3|a+b|+|a-b| is [numerical value].

If Σᵢ₌₁ⁿ(xᵢ-a)=n and Σᵢ₌₁ⁿ(xᵢ-a)²=na, (n,a>1) then the standard deviation of n observations x₁,x₂,.,xₙ is:

Let f(x) = 3sin?x + 10sin³x + 6sin²x - 3, x ∈ [-π/6, π/2]. Then, f is

If $\mathrm{c}$ $$ is a point at which Rolle's theorem holds for the function, ${}_{}\left(\frac{{}^{}}{}\right)$ $f\left(x\right)={\mathrm{l}\mathrm{o}\mathrm{g}}_e?\left(\frac{x^2+\alpha }{7x}\right)$ in the interval $\left[\mathrm{3,4}\right]$ $$ , where $$ , $\alpha \in R$ then ${}^{}$ $f^{\mathrm{\text{'}}\mathrm{\text{'}}}\left(c\right)$ is equal to:

If the lines x+y=a and x-y=b touch the curve y=x²-3x+2 at the points where the curve intersects the x-axis, then a/b is equal to

If x=1 is a critical point of the function f(x)=(3x²+ax-2-a)eˣ, then