Questions on the evaluation of derivatives or problems involving derivatives (for example, use of the mean value theorem).

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Find real parametar $a,b,c$ such that function $f$ become convex function $f(x) = \begin{cases}ax^2+bx+c,& x<0\\1 ,& x \ge 0\end{cases}$

Find real parametar $a,b,c$ such that function $f$ become convex function $$f(x) = \begin{cases}ax^2+bx+c,& x<0\\1 ,& x \ge 0\end{cases}$$ My work: If $f(x)$ is convex function that means ...
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1answer
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zeros of two functions are alternate

Let $a,b,c,d$ be real numbers. Show that the zeros of the functions $f(x)=a\cos x+b\sin x$ and $g(x)=c\cos x+d\sin x$ are distinct and alternate whenever $ad-bc\neq 0$. Suppose $x_0\in \mathbb{R}$ ...
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4answers
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Horizontal tangent line of a parametric curve

Suppose $x=t^2,y=t^3$ is a parametric curve. Here's a quote from my textbook: The origin, which corresponds to $t=0$, is a singular point of the parametric curve, because $dx/dt=2t,dy/dt=3t^2$ are ...
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2answers
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Show that $\dfrac{\rm{d}^{L-m}}{\rm{d}x^{L-m}}\left(x^2-1\right)^L=\dfrac{(L-m)!}{(L+m)!}(x^2-1)^m\dfrac{\rm{d}^{L+m}}{\rm{d}x^{L+m}}(x^2-1)^L$

The question that follows is driving me insane as it forms part of a derivation of the Associated Legendre Functions Normalization Formula: ...
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0answers
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Integration and differentiation of Fourier series

I am interested in the properties of Fourier series under integration and differentiation, and I've noticed a "strange" phenomenon. Suppose I have a Fourier series which I Integrate, and suppose that ...